-T4-4-
THE THEORY OF THE
RIEMANN
ZETA-FUNCTION
BY
E. C. TITCHMARSH
F.R.S.
FORMERLY SAVILIAN PROFESSOR OF GEOMETRY IN THE
UNIVERSITY OF OXFORD
SECOND EDITION
REVISED BY
D. R. HEATH-BROWN
FELLOW OF MAGDALEN COLLEGE, UNIVERSITY OF OXFORD
021736
CLARENDON PRESS ¦ OXFORD

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) The Executors of the late Mrs K. Titchmarsh and D. R. Heath-Brown, 1986
First published 1951
Second edition 1986
Reprinted 1988
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British Library Cataloguing in Publication Data
Titchmarsh, E. C.
The theory of the Riemann zeta-function.
1. Calculus 2. Functions, Zeta
3. Riemann-Hilbert problems
I. Title
515.9'82 QA320
ISBN 0-19-853369-1
Library of Congress Cataloging in Publication Data
Titchmarsh, E. C. (Edward Charles), 1899-
The theory of the Riemann zeta-function.
Bibliography: p.
1. Functions, Zeta I. Heath-Brown, D. R.
II. Title.
QA246.T44 1986 512'.73 86-12520
ISBN 0-19-853369-1 (pbk.)
Printed in Great Britain by
Biddies Ltd.
Guildford and King's Lynn

PREFACE TO THE SECOND EDITION
Since the first edition was written, a vast amount of further work has
been done. This has been covered by the end-of-chapter notes. In most
instances, restrictions on space have prohibited the inclusion of full
proofs, but I have tried to give an indication of the methods used
wherever possible. (Proofs of quite a few of the recent results described
in the end of chapter notes may be found in the book by Ivic [3].) I have
also corrected a number of minor errors, and made a few other small
improvements to the text. A considerable number of recent references
have been added.
In preparing this work I have had help from Professors J. B. Conrey,
P. D. T. A. Elliott, A.Ghosh, S. M. Gonek, H.L.Montgomery, and
S. J. Patterson. It is a pleasure to record my thanks to them.
oxford D. R. H.-B.
1986

PREFACE TO FIRST EDITION
This book is a successor to my Cambridge Tract The Zeta-Function of
Riemann, 1930, which is now out of print and out of date. It seems no
longer practicable to give an account of the subject in such a small space
as a Cambridge Tract, so that the present work, though on exactly the
same lines as the previous one, is on a much larger scale. As before, I do
not discuss general prime-number theory, though it has been convenient
to include some theorems on primes.
Most of this book was compiled in the 1930's, when I was still
researching on the subject. It has been brought partly up to date by
including some of the work of A. Selberg and of Vinogradov, though a
great deal of recent work is scantily represented.
The manuscript has been read by Dr. S. H. Min and by Prof. D. B.
Sears, and my best thanks are due to them for correcting a large number
of mistakes. I must also thank Prof. F. V. Atkinson and Dr. T. M. Fleet
for their kind assistance in reading the proof-sheets.
oxford E.C.T.
1951

CONTENTS
I. THE FUNCTION ?(*) AND THE DIRICHLET SERIES RELATED
TO IT 1
Definition of ?(s). Dirichlet series connected with ?(s).
Sums involving ojji). Ramanujan's sums.
II. THE ANALYTIC CHARACTER OF ?(*), AND THE FUNCTIONAL
EQUATION 13
Analytic continuation. Functional equation. The functions x(s), ?(s), and
E(s). Values at negative integers. Siegel's proof of the functional
equation. Zeros and the Hadamard product. Hamburger's theorem.
Selberg's method, Eisenstein series and the Maass-Selberg formula.
Tate's thesis. The p-adic zeta functions.
III. THE THEOREM OF HADAMARD AND DE LA VALLEE POUSSIN,
AND ITS CONSEQUENCES 45
Methods of Hadamard, de la Vallee Poussin, and Ingham.
The prime number theorem. Quantitative zero-free regions and bounds
for C(s)~\ C'(s)/C(s). Elementary estimates. Sieve methods. Proof via the
Maass-Selberg formula.
IV. APPROXIMATE FORMULAE 71
Estimates for exponential integrals. The approximate functional
equation. The Riemann-Siegel formula. Weighted approximate func-
functional equations. The approximate functional equation for ?(sJ.
V. THE ORDER OF ?(*) IN THE CRITICAL STRIP 95
The function fi(a). Weyl's method. Van der Corput's method. An improved
zero-free region. Other methods for bounding n(%). The method of
exponent pairs. Multiple sums and further work on
VI. VINOGRADOV'S METHOD 119
Vinogradov's mean-value theorem. Vinogradov's zero-free region.
Application to exponent pairs. The Vinogradov-Korobov estimate.
VII. MEAN-VALUE THEOREMS 138
The mean square. The fourth-power moment. Higher moments.
Convexity bounds. Fractional moments. Weighted mean-values. The
error term in the formula for the mean square. Atkinson's formula.
The twelfth-power moment. More on fractional moments. Applications
of Kloosterman sums. The mean-value theorems of Iwaniec, and of
Deshouillers and Iwaniec.
VIII. Q-THEOREMS 184
Results for ?A -I- it) and 1/?A + it). The range \ ^ a < 1.
Sharper bounds.

CONTENTS
IX. THE GENERAL DISTRIBUTION OF ZEROS 210
The Riemann-von Mangoldt formula. The functions S(t) and S^t). Gaps
between ordinates of zeros. Estimates for N(o, T). Selberg's bound for
1
N(o, T) da. Mean-values of S(t) and Sj (t). More on the distribution of
2.
the ordinates of zeros. Sign changes and distribution of S(t). Further
work on N(o, T). Halasz' lemma and the Large Values Conjecture.
X. THE ZEROS ON THE CRITICAL LINE 254
Riemann's memoir. The existence of an infinity of zeros on a = \. The
function N0(T). The Hardy-Littlewood bound. Selberg's theorem.
Functions for which the Riemann hypothesis fails. Levinson's method.
Simple zeros. Zeros of derivatives <^(m)(s).
XI. THE GENERAL DISTRIBUTION OF VALUES OF ?(*) 292
Values taken by ?(<x -I- it). The case a > 1. The case ^ < a < 1.
Voronin's universality theorem. The distribution of log ?(A -I- it).
XII. DIVISOR PROBLEMS 312
Basic results. Estimates for ak. The case k = 2. Mean-value estimates and
Q-results. Large values of k. More on the case k = 2. More Q-results.
Additional mean-value theorems.
XIII. THE LINDELOF HYPOTHESIS 328
Necessary and sufficient conditions. Mean-value theorems. Divisor
problems. The functions S(t), S^t), and the distribution of zeros.
XIV. CONSEQUENCES OF THE RIEMANN HYPOTHESIS 336
Deduction of the Lindelof hypothesis. The function v(<x).
Sharper bounds for ?(s). The case a = 1. The functions S(t) and S^t).
Bounds for ?(s) with a near ^. Mean-value theorems for S(t) and S^t). The
function M(x). The Mertens hypothesis. Necessary and sufficient con-
conditions for the Riemann hypothesis. More on S(t), S^t), and gaps between
zeros. Montgomery's pair-correlation conjecture. Disproof of the
Mertens conjecture, and of Turan's hypothesis.
XV. CALCULATIONS RELATING TO THE ZEROS 388
Location of the smallest non-trivial zero. Results of computer
calculations.
ORIGINAL PAPERS 392
FURTHER REFERENCES 406

-*¦ I
^ FUNCTION ?(s) AND THE DIRICHLET
-ij SERIES RELATED TO IT
1.1. Definition of ?(s). The Riemann zeta-function ?(s) has its origin in
the identity expressed by the two formulae
n=l
where n runs through all integers, and
where p runs through all primes. Either of these may be taken as the
definition of ?(.$); s is a complex variable, .9 = a-\~it. The Dirichlet series
A.1.1) is convergent for a > 1, and uniformly convergent in any finite
region in which a > 1-f-S, § > 0. It therefore defines an analytic func-
function ?(s), regular for a > 1.
The infinite product is also absolutely convergent for a > 1; for so is
1
pS
this being merely a selection of terms from the series ]? n~a. If we
expand the factor involving p in powers of p~s, we obtain
On multiplying formally, we obtain the series A.1.1), since each
integer n can be expressed as a product of prime-powers pm in just one
way. The identity of A.1.1) and A.1.2) is thus an analytic equivalent
of the theorem that the expression of an integer in prime factors is
unique.
A rigorous proof is easily constructed by taking first a finite number
of factors. Since we can multiply a finite number of absolutely con-
convergent series, we have
where nx, n2,..., are those integers none of whose prime factors exceed P.

THE FUNCTION ?(*) AND
Chap. I
Since all integers up to P are of this form, it follows that, if ?(s) is
denned by A.1.1),
This tends to 0 as P -> oo, if ct > 1; and A.1.2) follows.
This fundamental identity is due to Euler, and A.1.2) is known as
Euler's product. But Euler considered it for particular values of s only,
and it was Riemann who first considered ?(s) as an analytic function
of a complex variable.
Since a convergent infinite product of non-zero factors is not zero,
we deduce that ?(s) has no zeros for a > 1. This may be proved
directly as follows. We have for a > 1
where mx, ra2,..., are the integers all of whose prime factors exceed P.
Hence
'i_!Wi_i_W l x
if P is large enough. Hence |?(s)| > 0.
The importance of ?(s) in the theory of prime numbers lies in the
fact that it combines two expressions, one of which contains the primes
explicitly, while the other does not. The theory of primes is largely
concerned with the function tt{x), the number of primes not exceeding x.
We can transform A.1.2) into a relation between ?(s) and tt(x)', for if
a > 1,
-^) = - 2 M^)-^-
p
n + 1 oo
f l—dx = s f 7r{X
J *(:*¦-1) J ajC"-
\
dx.
A.1.3)
n
The rearrangement of the series is justified since 7r(n) < n and
—n~s) = 0(n"CT).

1.1 THE DIRICHLET SERIES
wrH)
and on carrying out the multiplication we obtain
where /x(l) = 1, /x(n) = ( — 1)* if n is the product of & different primes,
and fi(n) = 0 if n contains any factor to a power higher than the first.
The process is easily justified as in the case of ?(s).
The function n(n) is known as the Mobius function. It has the
2 ()
d\q
where d \ q means that d is a divisor of q. This follows from the identity
00 , 00 , v 00 ,
ns q
m=l n = l g = l
It also gives the 'Mobius inversion formula'
d|3
connecting two functions/(n), g(n) defined for integral n. If/ is given
and g defined by A.1.6), the right-hand side of A.1.7) is
d|g v ' r\d
The coefficient of f(q) is fi(l) = 1. If r < q, then d = kr, where k \ q/r.
Hence the coefficient of /(r) is
2 1 Q \ v \
^* I ^V /*^ ^ * i/** \fV /t*
by A.1.5). This proves A.1.7). Conversely, if g is given, and/is defined
by A.1.7), then the right-hand side of A.1.6) is
and this is g(q), by a similar argument. The formula may also be

4 THE FUNCTION C(s) AND Chap. I
derived formally from the obviously equivalent relations
where
71=1
Again, on taking logarithms and differentiating A.1.2), we obtain, for
p
p
\ A.1.8)
where A(n) = \ogp if n is p or a power of p, and otherwise A(n) = 0.
On integrating we obtain
where Ax(n) = A(n)/log n, and the value of log ?(s) is that which tends
to 0 as a -> oo, for any fixed t.
1.2. Various Dirichlet series connected with ?(s). In the first
place
where d(n) denotes the number of divisors of n (including 1 and n itself).
For =o oo oo
i
and the number of terms in the last sum is d(n). And generally
— (o- > 1), A-2.2)
ns
where 1c = 2, 3, 4,..., and dk(n) denotes the number of ways of expressing
n as a product of k factors, expressions with the same factors in a
different order being counted as different. For
CO 00 00
vi = l L v* = l * n=l vi...v*=n
and the last sum is dk(n).

1.2 THE DIRICHLET SERIES
Since we have also
on comparing the coefficients in A.2.1) and A.2.3) we verify the
elementary formula
d(n) = (m1 + l)...(mr+l) A.2.4)
for the number of divisors of
n = p™*p™:..p?r. A.2.5)
Similarly from A.2.2)
Aiyr- (L2-6)
We next note the expansions
where fj.(n) is the coefficient in A.1.4);
where v(w) is the number of different prime factors of n;
To prove A.2.7), we have
and this differs from the formula for l/?(s) only in the fact that the
signs are all positive. The result is therefore clear. To prove A.2.8), we
have ?w =r\ l-p~2s =r\1+p~s
?Bs) 1 1 A— p~sJ 1 1 1— p~s
p v
v

6 THE FUNCTION C(s) AND Chap. I
and the result follows. To prove A.2.9),
-FT l~p~2s = TT l+
11 (l-p-3K 1 1 {l-
?Bs) 11 (l-p-3K 1 1 {l-p->J
p
p
and the result follows, since, if n is A.2.5),
1(
Similarly
l-p-sL 1 1 (l-p-s
(p)
= TI
and A.2.10) follows.
Other formulae are
where X(n) = (—-l)r if n has r prime factors, a factor of degree k being
counted k times;
^fM (a>2), A.2.12)
n
where cf>(n) is the number of numbers less than n and prime to n; and
00
71 = 1
where a(n) is the greatest odd divisor of n. Of these, A.2.11) follows
at once from
^(s) 1 1 \1—Jp~2aJ 1 1 \1+P~7 J- J-
Also

1.2 THE DIRICHLET SERIES
and A.2.12) follows, since, if n = p?1...p?r,
Finally
1-21-* ,4 1-21-*
v
1 1 1
1 — 2-* 1 — 31-* 1—51-3'"
/, . 1 . 1 . \/, . 3 . 32
and A.2.13) follows.
Many of these formulae are, of course, simply particular cases of the
general formula
1+
tc = 1 p x -f Jr I
where f(n) is a multiplicative function, i.e. is such that, if n =
Again, let fk(n) denote the number of representations of n as a product
of k factors, each greater than unity when n > 1, the order of the factors
being essential. Then clearly
(a > 1). A.2.14)
71 = 2 fi
Let f(n) be tlie number of representations of n as a product of factors
greater than unity, representations with factors in a different order
being considered as distinct; and let/(I) = 1. Then
/(*) =kt/k{n)'
Hence m „. . „, .
A.2.15)
It is easily seen that ?(s) = 2 for s = a, where a is a real number greater
than 1; and |?(s)| < 2 for a > a, so that A.2.15) holds for a > a.

8 THE FUNCTION C(s) AND Chap. I
1.3. Sums involving <?a(n). Let aa(n) denote the sum of the ath
powers of the divisors of n. Then
00 , 00 00
Z? 2
V=l 71=1
i.e. ?(«K(«-a) = JT ^1 (a>l, a>R(a)+l). A.3.1)
n=l
Since the left-hand side is, if a =? 0,
we have aa{n) = ri ... Fr , A.3.2)
if n is A.2.5), as is also obvious from elementary considerations.
The formulaf
?—a—6) Zl/ ws
' 71=1
is valid for o > max{l,R(a)+l,RF)+l,R(a+6) + l}. The left-hand
side is equal to
I p-2s+a+b
1 1 n—q)s)(l—
Putting p~s = z, the partial-fraction formula gives
1— pa+bz2
A—«)A— paz)(l— pbz)(l— pa+bz)
pb
(l—pa)(l—pb)\l—z l—paz \—pbz'l—pa+bz
(I «(«+Da)(l—v(m+l)b\zm
| Bamanujan B), B. M. Wilson A).

1.3 THE DIRICHLET SERIES
Hence
U2s—a—b) v lv v
p m=0 ^ r r
and the result follows from A.3.2). If a = 6 = 0, A.3.3) reduces to
A.2.10).
Similar formulae involving cr^in), the sum of the ath powers of those
divisors of n which are qth. powers of integers, have been given by
Crum A).
1.4. It is also easily seen that, if/(w) is multiplicative, and
Zj ns
71 = 1
is a product of zeta-functions such as occurs in the above formulae, and
k is a given positive integer, then
)
Z* ns
71 = 1
can also be summed. An example will illustrate this point. The function
ora(n) is 'multiplicative', i.e. if m is prime to n
aa(mn) = ara(m)cra(n).
Hence
ns 1 I Z, p
71 = 1 P «l=0
and, if k = JJ pl,
2°° ctjjen) = -r-r y aa(pl+m)
ns I 1 2-i p™3
Hence
2
p
71=1 P 7/1 = 0
^uii^iri
Now if a 7^ 0,
1
Hence
, p A— p^p™ ~ A — pa)(l — J9"s)( 1 — pa~s)
Making a -> 0, ^ ^^ = ^E) U (^+!-^-s)- (!-4-2)
71 = 0
ws 3|X

10 THE FUNCTION f(«) AND Chap. I
1.5. Ramanujan's sums.f Let
ck(n) = 2 6-«»^/* = X cos~, A.5.1)
h ?-i K
h
where h runs through all positive integers less than and prime to k.
Many formulae involving these sums were proved by Ramanujan.
We shall first prove that
(n)= V J^jd. A.5.2)
<Z|*d|n * '
ck(n)=
<Z|*.d|n
k-1
The sum -qk(n) = 2 e~2nmnilk
is equal to k if k \ n and 0 otherwise. Denoting by (r, d) the highest
common factor of r and d, so that (r, d) = 1 means that r is prime to d,
d\k d\
Hence by the inversion formula of Mobius A.1.7)
(k\
4 \CLI
d\k x '
and A.5.2) follows. In particular
c*U) = /*(*)• A.5-3)
The result can also be written
tt
Hence
2
dr—k,d\n
ck(n)
k8 Zw r3
dr= k,d\n
Summing with respect to k, we remove the restriction on r, which now
assumes all positive integral values. HenceJ
ck(n) _ y H(r)dl_s _ vx-s{n) ,. - ..
r,d\n bV '
the series being absolutely convergent for a > 1 since |cA.(^) | ^ crx{n),
by A.5.2).
We have also
y *\d
, ns Z* ^\d
71=1 d\kd\n
= v J-) d y _JL = ?(S) y J?\ di-s. (i.5.5)
Z, ^\d A, (md)s K ' /-, ^\d v '
d\k v ' m=lv ' d\k x '
t Ramanujan C), Hardy E).
+ Two more proofs are given by Hardy, Ramanujan, 137-41.

1-5 THE DIRICHLET SERIES 11
We can also sum series of the forcnf
2~
ns
where f(n) is a multiplicative function. For example,
^A ck(n)d(n) ^ d(n) ^
Z ~~n* = Z ~tf~ Z
71 = 1 71=1 8|A:,8|n
^[s Z
8 X ;
m=l
VJJ J
8\k ^ ' P\h
if 8 = 17 pl. If k = Yl Px the sum is
n
V'%V,V
Hence
Mtr i ^T ck(Qn)f(n)
We can also sum > K* J '.
71 = 1
For example, in the simplest case f(n) = 1, the series is
V l V * (k\
7i = l h\k,h\qn
For given 8, n runs through those multiples of S/q which are integers.
If hjq in its lowest terms is BJq^ these are the numbers 8V 281}....
Hence the sum is
8\k
t Crum A).

12 THE FUNCTION f(s) AND Chap. I
Since 8X = B/(q,B), the result is
^CM^gW A.5.7)
71 = 1 51* * '
1.6. There is another class of identities involving infinite series of
zeta-functions. The simplest of these isf
p n = l
i
We have log?(«) = V V _i- = V
where P(s) = 2#~s. Hence
y Slogans) = y
^) y
n Z-i m , r <
71 = 1 771=1 r=l 7l|r
and the result follows from A.1.5).
A closely related formula is
00 / \ 00
2 1? = m 2 ^S^), A.6.2)
71 = 1 71 = 1
where v(n) is defined under A.2.8). This follows at once from A.6.1) and
the identity « „
ws ~ 2-, ms 2-, Vs'
71 = 1 7/1 = 1 3>
Denoting by 6(w) the number of divisors of n which are primes or
powers of primes, another identity of the same class is
71=1 71=1
where <f)(n) is defined under A.2.12). For the left-hand side is equal to
V — V l~ — — \
V
and the series on the right is
*i _L_ = y y
Z Z
yy y y -L y
71 Zy Zv mpmnS Zy Zy V»VS Zy
71 = 1 7M = 1 P P V n\v
Since
the result follows.
f See Landau and Walfisz A), Estermann A), B).

II
THE ANALYTIC CHARACTER OF J(*), AND
THE FUNCTIONAL EQUATION
2.1. Analytic continuation and the functional equation, first
method. Each of the formulae of Chapter I is proved on the supposi-
supposition that the series or product concerned is absolutely convergent. In
each case this restricts the region where the formula is proved to be valid
to a half-plane. For ?(s) itself, and in all the fundamental formulae of
§ 1.1, this is the half-plane a > 1.
We have next to inquire whether the analytic function ?(s) can be
continued beyond this region. The result is
Theorem 2.1. The function ?(s) is regular for all values of s except
s = 1, where there is a simple pole with residue 1. It satisfies the functional
equation ^ = 2V»-isin?s7rr(l-s)?(l-s). B.1.1)
This can be proved in a considerable variety of different ways, some
of which will be given in later sections. We shall first give a proof
depending on the following summation formula.
Let (f)(x) be any function with a continuous derivative in the interval
[a, b]. Then, if [x] denotes the greatest integer not exceeding x,
b b
= f 4>(x) dx + f (x-[x]-W(x) dx+
a a
. B.1.2)
Since the formula is plainly additive with respect to the interval (a, b ]
it suffices to suppose that n ^ a < b ^ n + 1. One then has
b r
Ux-n- \L>\x) dx = (b-n- JHF) - (a - n - JH(o) - <$>{x)dx,

14 ANALYTIC CHARACTER OF C(s) AND Chap. II
on integrating by parts. Thus the right hand side of B.1.2) reduces
to ([b]-n)(f)(b). This vanishes unless b = n + l, in which case it is
(f)(n + 1), as required.
In particular, let <f>(n) = n~s, where s ^ 1, and let a and 6 be positive
integers. Then
First take a > 1, a = 1, and make b ->• oo. Adding 1 to each side, we
obtain
00
Since [#]—x-\-\ is bounded, this integral is convergent for a > 0, and
uniformly convergent in any finite region to the right of a = 0. It
therefore defines an analytic function of s, regular for o > 0. The
right-hand side therefore provides the analytic continuation of ?(s) up
to a = 0, and there is clearly a simple pole at s = 1 with residue 1.
For 0 < a < 1 we have
f W—*j f , j ! s f°^ 1
L J , dx = — x~s dx = , - —- = -,
J z^1 J s— 1' 2J Xs*1 2'
0 0 1
and B.1.4) may be written
00
J
B<L5)
o
Actually B.1.4) gives the analytic continuation of ?(s) for a > —1;
for if
x
f(x) = [x]-x+l, A(x) = jf(y) dy,
i
then fx{x) is also bounded, since, as is easily seen,
f f(y) dy = O
k
for any integer k. Hence
J

2.1 THE FUNCTIONAL EQUATION 15
which tends to 0 as xx -> oo, x2 -> oo, if a > — 1. Hence the integral in
B.1.4) is convergent for a > — 1. Also it is easily verified that
f [x\—x+\ , 11,
S L_J J^dx = 1— (cr < 0).
J xs+1 s-1^2 v ;
0
00
Hence
d
Now we have the Fourier series
r i ii "ST sin2?i7ra; /n , wv
[x]—x+$= > , 2.1.7
^—t nrr
n=l
where x is not an integer. Substituting in B.1.6), and integrating
term by term, we obtain
dx
= s f fet^±i<?r (-1< a < 0). B.1.6)
00
«v^ f
-y -
rr ?-1 71 J
fsi
J
n J
n-l Q
= -Btt
7T
i.e. B.1.1). This is valid primarily for —1 < a < 0. Here, however,
the right-hand side is analytic for all values of s such that a < 0. It
therefore provides the analytic continuation of ?(s) over the remainder
of the plane, and there are no singularities other than the pole already
encountered at s = 1.
We have still to justify the term-by-term integration. Since the
series B.1.7) is boundedly convergent, term-by-term integration over
any finite range is permissible. It is therefore sufficient to prove that
,. ^ 1 f sin2tt.7ra; ,
lim > - ;—dx = 0 ( — 1 < a < 0).
n-l ^
OO 00
,T fsin2?i7ra; , [ cos 2mrx~\ °° s+1 rcos2?i7ra; ,
Now j— dx = j- ¦— -T77L— dx
U J
A
- o
and the desired result clearly follows.

16 ANALYTIC CHARACTER OF C(s) AND Chap. II
The functional equation B.1.1) may be written in a number of
different ways. Changing s into 1—s, it is
?(l-s) = 21-S7r-Scos^7rr(s)^(s). B.1.8)
It may also be written
W K«), B.1.9)
where x(s) = 2s7rs-1sin?s7rr(l — s) = Wr^~^, B.1.10)
and X(*)x(l~*) = 1- B.1.11)
Writing $(s) = %s(s-lOr-isr(%s)i;{s), B.1.12)
it is at once verified from B.1.8) and B.1.9) that
?(s) = $A-8). B.1.13)
Writing S(z) = |(?+iz) B.1.14)
we obtain S(z) = E(—z). B.1.15)
The functional equation is therefore equivalent to the statement that
E(z) is an even function of z.
The approximation near s = 1 can be carried a stage farther; we have
o— J.
where y is Euler's constant. For by B.1.4)
= lim f
n
r \x~\-x
LXJ x
X2
dx+1
— -logn+1
= Km
= lim| ^>
2.2. A considerable number of variants of the above proof of the
functional equation have been given. A similar argument was applied
by Hardy,f not to ?(s) itself, but to the function
n=l
Hardy F).

2.2 THE FUNCTIONAL EQUATION 17
This Dirichlet series is convergent for all real positive values of s, and
so, by a general theorem on the convergence of Dirichlet series, for all
values of s such that a > 0. Here, of course, the pole of ?(s) at s = 1
is cancelled by the zero of the other factor. These facts enable us to
simplify the discussion in some respects.
Hardy's proof runs as follows. Let
Z
V
Z 2n+l '
n = 0
This series is boundedly convergent and
f(x) = (—1)«Jtt for rmr < x < (m+ljw (m = 0, 1,...).
Multiplying by Xs-1 @ < s < 1), and integrating over @, oo), we obtain
00 (m+lOT ^
(-1) I a^ rfo: = r(s)sin45tt V!
*L n=0
m=o ^ n=o
The term-by-term integration may be justified as in the previous proof.
The series on the left is
TO=1
This series is convergent for s < 1, and, as a little consideration of the
above argument shows, uniformly convergent for R(s) < 1—8 < 1.
Its sum is therefore an analytic function of s, regular for R(s) < 1.
But for s < 0 it is
Its sum is therefore the same analytic function of s for R(s) < 1.
Hence, for 0 < s < 1,
(l
2s
and the functional equation again follows.
2.3. Still another proof is based on Poisson's summation formula
00
2 f(n) = 2 f f(u)coa 27mu du. B.3.1)
= CO OO •
n=-co -oo
If we put/(a;) = \x\~s and ignore all questions of convergence, we obtain
the result formally at once. The proof may be established in various
ways. If we integrate by parts to obtain integrals involving sin 2nnu,

18 ANALYTIC CHARACTER OF C(s) AND Chap. II
we obtain a proof not fundamentally distinct from the first proof given
here.f The formula can also be used to give a proof depending! on
Actually cases of Poisson's formula enter into several of the following
proofs; B.6.3) and B.8.2) are both cases of Poisson's formula.
2.4. Second method. The whole theory can be developed in another
way, which is one of Riemann's methods. Here the fundamental
formula is x
o
To prove this, we have for a > 0
00 00
f &-H-™ dx = — I y*-H-v dy
ns
0 0
Hence
00
r(«)?(«) = V (xs~1e-nxdx= f x3-1 y e~nx dx = f -^-
n-lj/ i n-l J 6X~1
if the inversion of the order of summation and integration can be
justified; and this is so by absolute convergence if a > 1, since
00
f f xa-H~™ dx = I»?(a)
is convergent for cr > 1.
Now consider the integral
|
c
where the contour C starts at infinity on the positive real axis, encircles
the origin once in the positive direction, excluding the points ±2irr,
±4171,..., and returns to positive infinity. Here zs~x is defined as
e(s-l)logz
when the logarithm is real at the beginning of the contour; thus I (log z)
varies from 0 to 2tt round the contour.
We can take C to consist of the real axis from oo to p @ < p < 2tt),
the circle \z\ = p, and the real axis from p to oo. On the circle,
|2s-l| _ e(a-l)log|2Marg2 < Igla-l^Trl^
|e*-l| >A\z\.
t Mordell B). % Ingham, Prime Numbers, 46.

2.4 THE FUNCTIONAL EQUATION 19
Hence the integral round this circle tends to zero with p if a > 1. On
making p -> 0 we therefore obtain
00
= (e2™-l)r(s)?(s)
_ 2irrei7TS
-f(I37LE)-
Hence ?(s) = e \~v- f 4~~-dz. B.4.2)
2tti J ez—1
c
This formula has been proved for a- > 1. The integral /(s), however,
is uniformly convergent in any finite region of the s-plane, and so defines
an integral function of s. Hence the formula provides the analytic
continuation of ?(s) over the whole s-plane. The only possible singu-
singularities are the poles of F(l—s), viz. s = 1, 2, 3,.... We know already
that ?(s) is regular at s = 2, 3,..., and in fact it follows at once from.
Cauchy's theorem that I(s) vanishes at these points. Hence the only
possible singularity is a simple pole at s = 1. Here
and T(l—s)= — + ....
s—1
Hence the residue at the pole is 1.
If s is any integer, the integrand in I(s) is one-valued, and /(s) can
be evaluated by the theorem of residues. Since
where B1} B2,--. are Bernoulli's numbers, we find the following values
of Us):
?@) = -i, ?(-2m) = 0, l>(\-2m) = K ^jg" (m = 1, 2,...).
B.4.3)
To deduce the functional equation from B.4.2), take the integral
along the contour Cn consisting of the positive real axis from infinity
to Bn-\-l)rr, then round the square with corners B71+1O1(^1^1), and
then back to infinity along the positive real axis. Between the contours

20 ANALYTIC CHARACTER OF f(s) AND Chap. II
C and Cn the integrand has poles at the points ±2irr,..., ±2imr. The
residues at 2mi-n and — 2mi7r are together
= — 2BmTr)s-1eiiTS sin \tts.
Hence by the theorem of residues
C
2 Bm7r)s-1.
J m=l
Cn
Now let a < 0 and make n->co. The function l/(es— 1) is bounded
on the contours Cn, and z3 = O^z^-1). Hence the integral round Cn
tends to zero, and we obtain
00
sin \tts 2 BK1
m=l
— s).
The functional equation now follows again.
Two minor consequences of the functional equation may be noted
here. The formula
?Bm) = 22'"-17r2w-^T (m = 1, 2,...) B.4.4)
follows from the functional equation B.1.1), with s = 1 —2m, and the
value obtained above for ?A—2m). Also
r@) = —|log27r. B.4.5)
For the functional equation gives
In the neighbourhood of s = 1
and t'W
?()
where A; is a constant. Hence, making s -> 1, we obtain
-88 ->•-
and B.4.5) follows.
2.5. Validity of B.2.1) for all 5. The original series A.1.1) is naturally
valid for a > 1 only, on account of the pole at s = 1. The series B.2.1)
is convergent, and represents A — 21~s)?(s), for a > 0. This series ceases

2.5 THE FUNCTIONAL EQUATION 21
to converge on a = 0, but there is nothing in the nature of the function
represented to account for this. In fact if we use summability instead
of ordinary convergence the equation still holds to the left of a = 0.
00
Theorem 2.5. The series 2 (—l)'1"^ is summable (A) to the sum
n = l
A — 21~s)?(s) for all values of s.
Let 0 < x < 1. Then
J^ ^ —\\n-lxn f
Z-, n3 Zw T(s) J
n=l n=l v ' J
n=l n=l
00
00 oo °°
T(s)J Z* v r(«)J l+a;e-«
o n-1 o
This is justified by absolute convergence for a > 1, and the result by
analytic continuation for a > 0.
We can now replace this by a loop-integral in the same way as
B.4.2) was obtained from B.4.1). We obtain
ns 2m J \+xe
IVs-1
J \+
xe~w
,-w
dw,
when C encircles the origin as before, but excludes all zeros of l-\-xe~w,
i.e. the points w = Iogx-j-Bra+l)i7r.
It is clear that, as x -> 1, the right-hand side tends to a limit, uniformly
in any finite region of the s-plane excluding positive integers; and, by
the theory of analytic continuation, the limit must be A — 21~s)?(s).
This proves the theorem except if s is a positive integer, when the
proof is elementary.
Similar results hold for other methods of summation.
2.6. Third method. This is also one of Riemann's original proofs.
We observe that if a > 0
00
f xl3-^-11*™ dx =
o
Hence if a > 1
00 " 7 . 00
JS
= f f afr-ie-n*"* dx= 1 xh-i \ e-nt™ dx,
n-ljj I —1
jj
the inversion being justified by absolute convergence, as in § 2.4.

22 ANALYTIC CHARACTER OF ?(s) AND Chap. II
Writing «,
0(*O = 2 e~n2™ B.6.1)
we therefore have „
«) = j~j J **-V(«) d* (o- > !)• B.6.2)
Now it is known that, for x > 0,
00 1
V g-n2nx __
7l=_oo \'x
{^) ). B.6.3)
Hence B.6.2) gives
1 CO
= f a^'-V^) dx+ \ ^-^{x) dx
0 1
1
= J
i- i+ fo;is-Wij^+ [xt-hf,(x)dx
s—
0
oo
= _l + f
i
The last integral is convergent for all values of s, and so the formula
holds, by analytic continuation, for all values of s. Now the right-hand
side is unchanged if s is replaced by 1 — s. Hence
ir-*T(\8)Z(8) = 7r-f4T(J-WS(l-«), B.6.4)
which is a form of the functional equation.
2.7. Fourth method; proof by self-reciprocal functions. Still
another proof of the functional equation is as follows. For a > 1,
B.4.1) may be written
= (Ul
J \ex-
-l x) ^s-l^J ex-l '
o i
and this holds by analytic continuation for a > 0. Also for 0 < a < 1
00
1 _ _ far5-1
s—\ J x
dx.

2.7 THE FUNCTIONAL EQUATION 23
CO
Hence ?(*)]» = J f-i_ -^W <& @ < a < 1). B.7.1)
o
Now it is known that the function
<2-7-2>
is self-reciprocal for sine transforms, i.e. that
00
/(*) = Jir\ (f(y)smxy dy. B.7.3)
o
Hence, putting x = ?<JBtt) in B.7.1),
0
oo oo
o o
If we can invert the order of integration, this is
00 00
2\s+\tt\s-\ \f{y) dy J ^si
0 0
oo
t?*-? r f{y)y~3 dy \ ws-1si
J
.) 2cosiJrA_5),
and the functional equation again follows.
To justify the inversion, we observe that the integral
00
J f(y)sin?ydy
o
converges uniformly over 0 < 8 ^ ? < A. Hence the inversion of this
part is valid, and it is sufficient to prove that
lim J/(y) dylj + j)?-1siii& d? = 0.
s
y-L sin.
o
Now J ?*-! si
0 0
and also = y~s \ u'^&uiudu = 0(y~a).

24 ANALYTIC CHARACTER OF C(s) AND Chap. II
Since f(y) = 0A) as y -> 0, and = O(y~1) as y -> oo, we obtain
00 8
0 0
1 1/8 00
= J 0(8a+1y) <&/+ J 0(8ff+1) dJy+ f Oiy-*-1) dy = 0(8") -> 0.
o l i/8
A similar method shows that the integral involving A also tends to 0.
2.8. Fifth method. The process by which B.7.1) was obtained from
B.4.1) can be extended indefinitely. For the next stage, B.7.1) gives
o i
and this holds by analytic continuation for ct > — 1. But
00 1
= -— (-1 <a < 0).
1
Hence
00
. B.8.1)
Now _L_ = i_i + 2a; V ——i -. B.8.2)
ex-l x 2^ Z,4n27r2+a;2 v
71 = 1
Hence
J 1
o
= 2
2 COS iSTT COS iS',
71 = 1 * *
the functional equation. The inversion is justified by absolute con-
convergence if — 1 < ct < 0.
2.9. Sixth method. The formulaf
c + ioo
C —loo
is easily proved by the calculus of residues if ct > 1; and the integrand
is 0{\z\-a-x), so that the integral is convergent, and the formula holds
by analytic continuation, if ct > 0.
j" Klooaterman A).

2.9 THE FUNCTIONAL EQUATION 25
We may next transform this into an integral along the positive real
axis after the manner of § 2.4. We obtain
00
ri , sinTrs f (T'(l+x) , )
0 B.9.2)
To deduce the functional equation, we observe thatf
00
T(x) ° 2x
o
Hence
_2 f tdt
x J (t2+x2){e™-l)'
= ±_2 f tdt =_2 f
2x J (<2+a;2)(e2^-l) J
o
Hence B.9.2) gives
- ^ J a;
00 00
-s ^ i J__/* __M dt
j
0 0
0
cos ^7ts
0
f/_J L
J \e27Tt— 1 27t«
oo
= 2sini7rsB7r)s-1 f (— -)u~s du
2 J \e«-l w/
by B.7.1). The inversion is justified by absolute convergence.
2.10. Seventh method. Still another method of dealing with
due to Riemann, has been carried out in detail by Siegel.J It depends
on the evaluation of the following infinite integral.
Let <D(o)= ,„ , dw, B.10.1)
where L is a straight line inclined at an angle \n to the real axis, and
t Whittaker and Watson, § 12.32, example. J Siegel B).

26 ANALYTIC CHARACTER OF ?(s) AND Chap. II
intersecting the imaginary axis between 0 and 2-ni. The integral is
plainly convergent for all values of a.
We have
—eaw) dw
f
—<S>{a) =
J
— 1
L
= j e^iw2l7T+aw dw
L
__ I
L
(
where W = w—2i-na. Here we may move the contour to the parallel
line through the origin, so that the last integral is
00
eii7T f e-ip3ln dp = 2?reii7r.
Hence O(o+1)—0>(o) = 27re**a8+*>. B.10.2)
Next let L' be the line parallel to L and intersecting the imaginary
axis at a distance 2n below its intersection with L. Then by the theorem
of residues
dw—
J ew__X J ew_l
dw = 2ni.
J J ew_l
L' L
But
dw
f — dw = I
J e»-1 J
ew—
L'
-j
,^iw2Itt +w-iTT+a(w-27ri)
dw = —q-2tti<
Hence —e-2^«O(a+l)—$(a) = 27rt. B.10.3)
Eliminating O(a+1)> we have
27rt+2^ , B.W.4)
or *(a) = 27r°o^(K-a-i)ei^.-|,, BJ05)
cos?ra

2.10 THE FUNCTIONAL EQUATION 27
If a = \iz\-n-\-\, the result B.10.4) takes the form
J c»_1
dw = 2vi-
ez—l
L
Multiplying by 2s (a > 1), and integrating from 0 to coe~iiiT, we obtain
ooe-if7T
I ^ J I iiVl/l/ 1 7
J c»-l J
0 ooe~i'7T
= 27rir(s)^(s) — 2-ni zs~x dz.
J ez—l
o
The inversion on the left-hand side is justified by absolute convergence;
in fact i. !.
w = —c+pei177, 2 = re~ilTT,
where c > 0, so that K(izw) = —cr/^2.
Now
OOe~l*7T 00 / \ _a
0 0 V '
^-—J-zs-xdz = — — 6— -J-zs~1dz,
J ez— 1 l+e-^J ez—1
and
o Z
where 1/ is the reflection of L in the real axis. Hence
1
or
= ei^s-1Js-17ris-1r(is) f ^J
ii f e-iiz2ln+i2
is-|r(i —15) r__ 2s-1 ^2.
B.10.6)
This formula holds by the theory of analytic continuation for all
values of s.
If s = \-\-it, the two terms on the right are conjugates. Hence
f(s) = 7r-?sr(!<s)?(<s) is real on a = |. Hence
/(«) =/("+*«) =/(l-cr+»O =f{\-a-it) =f(l-8),
the functional equation.

28 ANALYTIC CHARACTER OF f(s) AND Chap. II
2.11. A general formula involving ?(s). It was observed by
Miintzf that several of the formulae for ?(s) which we have obtained
are particular cases of a formula containing an arbitrary function.
We have formally
""¦' 00
oo ,r
dx
f Xs-1 JT F{nx) &=:f f xs-1F(nx)
y n=l n=li
0
00
= 2 h \ yS'1F{y) dy
«=i «
00
dy,
where F(x) is arbitrary; and the process is justifiable if F(x) is bounded
in any finite interval, and 0(x~a), where a > 1, as x -> oo. For then
00 °°
n=l
f
exists if 1 < a < a, and the inversion is justified.
Suppose next that F'(x) is continuous, bounded in any finite interval,
and 0{x~P), where /? > 1, as x -> oo. Then as x -> 0
f F(nx)~ f F(ux) du = x f F'(ux)(u—[u]) du
n L 0 0
II x oa
= x j 0A) du-\-x j 0{{ux)-P} du = 0A),
0 1/x
00
i.e. fF(nx) = - ( Flv) dv+O{l) = - + 0A),
71 — 1 •«' J -^
0
say. Hence
00
r v
J n = l
1 00
/• / oo ^\ /» J^U
#) rfa:,
= f a^-1/ f ^(na:) --) rfa: + -^- + f z* V
J U-i *i 5-1 J n4i
and the right-hand side is regular for a > 0 (except at 5 = 1). Also
for ct < 1 «,
_?_ = _c f x8-2 dx.
5—1 J
t Miintz A).

2.11 THE FUNCTIONAL EQUATION 29
Hence we have Miintz's formula
00
( ys~1F(y)dy= ( Xs-1 2 Jtyia:)-! ( F(v)dv\dx, B.11.1)
0 Vn=l )
valid for 0 < a < 1 if i^x) satisfies the above conditions.
If F{x) = e~x we obtain B.7.1); if F{x) = e-™2 we obtain a formula
equivalent to those of § 2.6; if F{x) = 1/A+z2) we obtain a formula
which is also obtained by combining B.4.1) with the functional equation.
If F(x) = a^sinTra; we obtain a formula equivalent to B.1.6), though
this F(x) does not satisfy our general conditions.
If F{x) = 1/A +xJ, we have
,, n-l
nr, .. , X
tj A— SOT
Hence ^——— i
o
and on integrating by parts we obtain B.9.2).
2.12. Zeros; factorization formulae.
Theorem 2.12. |(s) and S(z) are integral functions of order 1.
It follows from B.1.12) and what we have proved about
that ?(s) is regular for a > 0, (s— l)?(s) being regular at s = 1. Since
?(s) = |(l—,s), |(s) is also regular for a < 1. Hence |(s) is an integral
function.
Also
00
J e-"^* rftt < f e-^tti"-1 dtt = r(J(r) = 0{eAexl0ea) {a > 0),
o o
B.12.1)
and B.1.4) gives for a ^ \, \s~\\ > A,
(
klj ^| + 0(l) = 0(|5|). B.12.2)
Hence B.1.12) gives g(s) = 0(e^lsll0gls|) B.12.3)
for a ^ \, \s\ > ^4. By B.1.13) this holds for a ^ $ also. Hence ?(«)
is of order 1 at most. The order is exactly 1 since as s -> oo by real
values log?(s) '~ 2~s, log|(s) '~ \s\ogs.

30 ANALYTIC CHARACTER OF C(s) AND Chap. II
Hence also E(z) = O(eA^lo«^) {\z\ > A),
and H(z) is of order 1. But E(z) is an even function. Hence H(Vz) is
also an integral function, and is of order \. It therefore has an infinity
of zeros, whose exponent of convergence is \. Hence S(z) has an infinity
of zeros, whose exponent of convergence is 1. The same is therefore
true of ?(s). Let p1} p2,... be the zeros of ?(s).
We have already seen that ?(s) has no zeros for a > 1. It then follows
from the functional equation B.1.1) that ?(s) has no zeros for a < 0
except for simple zeros at s = —2, — 4, —6,...; for, in B.1.1), ?A— s)
has no zeros for a < 0, sin^STr has simple zeros at s = —2, —4,... only,
and F(l— s) has no zeros.
The zeros of ?(s) at —2, —4,..., are known as the 'trivial zeros'. They
do not correspond to zeros of ?(s), since in B.1.12) they are cancelled
by poles of r(^s). It therefore follows from B.1.12) that ?(s) has no
zeros for a > 1 or for a < 0. Its zeros pv p2,... therefore all lie in the
strip 0 < a < 1; and they are also zeros of ?(s), since s(s—l)F(|s) has
no zeros in the strip except that at s = 1, which is cancelled by the
pole of ?(s).
We have thus proved that ?(s) has an infinity of zeros px, p2,... in
the strip 0 < a ^ 1. Since
(l_2i-«K(«)=l-I + I-...>0 @<«<l) B.12.4)
and ?@) ^ 0, ?(sj has no zeros on the real axis between 0 and 1. The
zeros px, p2,... are therefore all complex.
The remainder of the theory is largely concerned with questions about
the position of these zeros. At this point we shall merely observe that
they are in conjugate pairs, since ?(s) is real on the real axis; and that,
if p is a zero, so is 1—p, by the functional equation, and hence so is
I —p. If p — j8+iy, then 1—p~ — 1— fi+iy. Hence the zeros either lie
on a = \, or occur in pairs symmetrical about this line.
Since ?(s) is an integral function of order 1, and ?@) = —?@) = ?,
Hadamard's factorization theorem gives, for all values of s,
B.12.5)
where b0 is a constant. Hence
B.12.6)

2.12 THE FUNCTIONAL EQUATION 31
where b = b0-\-\\og7T. Hence also
Making s -> 0, this gives
Ho) _ h i i 1 Hi)
Since ?'@)/?@) = Iog27r and P(l) = — y, it follows that
6 = log27r-l-?y. B.12.8)
2.13. In this sectionf we shall show that the only function which
satisfies the functional equation B.1.1), and has the same general charac-
characteristics as ?(s), is ?(s) itself.
Let G(s) be an integral function of finite order, P(s) a polynomial, and
f(s) = G(s)/P(s), and let
n
be absolutely convergent for a > 1. Let
i l-l8y-fr-*, B.13.2)
where
the series being absolutely convergent for <r < —a < 0. Thenf{s) =
where C is a constant.
We have, for x > 0,
2—ioo
oo 2 + t"°°
n = l «_„•
!—loo
oo
Also, by B.13.2),
2 +ioo
= 2~. J ^(i-^r^-^K
-?A-S)~—is
2 —ioo
We move the line of integration from cr = 2 to cr= — 1 — a. We observe
f Hamburger (l)-D), Siegel A).

32 ANALYTIC CHARACTER OF f(«) AND Chap. II
that/(s) is bounded on a = 2, and g(l—s) is bounded on a ~ — 1—«;
since
it follows that g{l—s) = 0{\t\\) on a = 2. We can therefore, by the
Phragmen-Lindelof principle, apply Cauchy's theorem, and obtain
— a—l + ioo
) = ^ J
J
— a — 1 — ioo
where Rx, B2,..., are the residues at the poles, say s1}..., sm. Thus
m m
where the Qv(logx) are polynomials in log«. Hence
yX ?* 27TI J
n—1 —a—l—i
00
00
Hence f ane-^2a; = -j- V bne-*n*lx+$Q(x).
Multiply by e-nl>x (t > 0), and integrate over @, oo). We obtain
oo oo , oo
n=l v ' ' n=l 0
0
and the last term is a sum of terms of the form
00
I xa log6z e-nt*x dx,
o
where the 6's are integers and R(a) > —1; i.e. it is a sum of terms of
the form
71 = 1
where #(?) is a sum of terms of the form
Now the series on the left is a meromorphic function, with poles at
±:in. But the function on the right is periodic, with period i. Hence
(by analytic continuation) so is the function on the left. Hence the
residues at hi and (&+l)i are equal, i.e. ak = ak+1 (k = 1, 2,...). Hence
ak = ax for all k, and the result follows.

2.14 THE FUNCTIONAL EQUATION 33
2.14. Some series involving ?(s). We havef
-l}-... B.14.1)
for all values of s. For the right-hand side is
1 ((8-1)8 1 (S-1)S(S+1) 1
1.2 n21 1.2.3
oo
= 1 LV __!__ L_5_z^
s—l^Un—lK-1 n3-1 ns
=?(8).
The inversion of the order of summation is justified for o- > 0 by the
convergence of
The series obtained is, however, convergent for all values of s.
Another formulaj which can be proved in a similar way is
also valid for all values of s.
Either of these formulae may be used to obtain the analytic con-
continuation of ?(s) over the whole plane.
2.15. Some applications of Mellin's inversion formulae.§
Mellin's inversion formulae connecting the two functions f(x) and $(s)
are «>
0 — iao
The simplest example is
/(*) = e-, 5(«) = T(8) (a > 0). B.15.2)
From B.4.1) we derive the pair
^ (a > 1), B.15.3)
t Landau, Handbuch, 272. t Ramaswami A).
§ See E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, §§ 1.5, 1.29,
2.1, 2.7, 3.17.

34 ANALYTIC CHARACTER OF ?(s) AND Chap. II
and from B.6.2) the pair
f(x) == ifi(x), #(s) = 7T-sF{s)?{2s) {a > l). B.15.4)
The inverse formulae are thus
cr + ioo
1 f 1
2tti J ex— 1 v '
cr—ioo
cr + t"oo
1
and
If
2?7i J ;
cr—ioo
Each of these can easily be proved directly by inserting the series for
?(8) and integrating term-by-term, using B.15.2).
As another example, B.9.2), with 5 replaced by 1—s, gives the Mellin
pair
— @ <a < 1). B.15.7)
The inverse formula is thus
i T^fi,ds. B.15.8)
T(l+x) & 2% J sin775
Integrating with respect to x, and replacing s by 1—s, we obtain
cr+ioo
logr(l+z) — x\ogx-\rx = — — I J*?r!—ds @<a<l). B.15.9)
2i J s sin 77«s
cr—ioo
This formula is used by Whittaker and Watson to obtain the asymp-
asymptotic expansion of log r(l+a;).
Next, let f(x) and $(s) be related by B.15.1), and let g(x) and E(s)
be similarly related. Then we have, subject to appropriate conditions,
—. f %(s)E(w—8)d8=[f{z)g{z)zv>-1dz. B.15.10)
2tti J J
C — ioo 0
Take for example $(<s) = ©(«) = r(s)?(s), so that
Then, if R(zv) > 2, the right-hand side is
oo
-± ~2dx =
o o

2.15 THE FUNCTIONAL EQUATION 35
Thus if 1 <c<R{w)—1
C+iao
~ J Y{s)Y{w-s)l{s)l{w-s)ds = T{w)tt{w-\)~l{w)}. B.15.11)
C—loo
Similarly, taking %{s) = E(s) = T(s)?Bs), so that
f{x) = g(x) = Mx/tt) = |
71 = 1
the right-hand side of B.15.10) is, if R{w) > 1,
00 oo oo oo oo
This may also be written
where r(n) is the number of ways of expressing n as the sum of two
squares; or as T{w){tt{w)-n{w) — tt{2w)}y
where rj(w) = l-w—3-w+5-w—....
Hencef if | < c < R{w)~%
~ J
J
c-ico B.15.12)
2.16. Some integrals involving H(?). There are some cases % in
which integrals of the form
00
ar) = j f{t)Z{t) cos xtdt
o
can be evaluated. Let/(O = \<f>(it)\2 = <f>{it)<f>(—it), where <j> is analytic.
Writing y = ex,
c .
X) === ¦5 I vi^tjfpi—itfUtytyy dt
—00
00
= J J <t>{it)<t>{-it)Z{\+it)yil dt
= 2^ J #(«-
217" J #(«~W(i-*)(«-
J-XOO
t Hardy D). A generalization is given by Taylor A). % Ramanujan A).

36 ANALYTIC CHARACTER OF C(s) AND Chap. II
Taking <f>(s) — 1, this is equal to
1VJ c
y f
n x 2-ioo
, 3 + ioo 2 + ioo
1^1 |*
1 3-ioo ' ' 2-loo
4tt ^ / y \ .,_,.., 6tt ^ / v \~2
Hence
00
f S(Ocosa^ dt = 2tt2 § B7T7i4e-9a;/2-37i2e-5a:/2)exp(-%27re-2a;).
o n=1
B.16.1)
Again, putting <f>{s) = 1/E+1), we have
?-100
in the notation of § 2.6. Hence
•tdt = \-n{ehx—2e-2xifj(e-2x)}. B.16.2)
J
o
The case <?(s) = T(^s—^) was also investigated by Ramanujan, the
result being expressed in terms of another integral.
2.17. The function ?(s, a). A function which is in a sense a generali-
generalization of ((s) is the Hurwitz zeta-function, defined by
71 = 0
This reduces to ?(s) when a = 1, and to Bs— l)?(s) when a — \. We
shall obtain here its analytic continuation and functional equation,
which are required later. This function, however, has no Euler pro-
product unless a = % or a = 1, and so does not share the most characteristic
properties of ?(s).

2.17 THE FUNCTIONAL EQUATION 37
As in § 2.4
C(s, a) =fJL h-ie-in^ dx = 1 fW
n4*0r(*)J T(s)J 1-e-*
0 ° B.17.1)
We can transform this into a loop integral as before. We obtain
fi^ B'17-2)
c
This provides the analytic continuation of ?(s,a) over the whole plane;
it is regular everywhere except for a simple pole at s = 1 with residue 1.
Expanding the loop to infinity as before, the residues at 2m-ni and
— 2m7rt are together
Hence, if a < 0,
~^) f • i X~* cos 2m7T(i .. ^-*
—< sin tjttts y \- cos \tts y
x ' V m=l wi=l
B.17.3)
If a = 1, this reduces to the functional equation for ?(s).
NOTES FOR CHAPTER 2
2.18. Selberg [1] has given a very general method for obtaining the
analytic continuation and functional equation of certain types of zeta-
function which arise as the 'constant terms' of Eisenstein series. We
sketch a form of the argument in the classical case. Let ^f = {z = x + iy:
y > 0} be the upper half plane and define
(c d)= 1
and
B(z, s) = CBs)?(z, s) = ? f\
(c d)^@. 0)
these series being absolutely and uniformly convergent in any compact
subset of the region R(s) > 1. Here E{z, s) is an Eisenstein series, while
B(z, s) is, apart from the factor ys, the Epstein zeta-function for the
lattice generated by 1 and z. We shall find it convenient to work with
B(z, s) in preference to E(z, s).

38 ANALYTIC CHARACTER OF ?(s) AND Chap. II
We begin with two basic observations. Firstly one trivially has
B(z + l,s) = B(-1/2, s) = B(z, s). B.18.1)
(Thus, in fact, B(z, s) is invariant under the full modular group.)
Secondly, if A is the Laplace-Beltrami operator
y \dx* dy*)'
then
whence
AB(z, s) = s(l-s)B(z, s) {a > 1). B.18.3)
We proceed to obtain the Fourier expansion of B{z, s) with respect to x.
We have
B(z,s)= ? an(y,s)e2*inx,
_ 00
where
r e-2ninxdx
J \cx + d + icy\2s
o
1
r e-2KinX(lx
0
with dn = 1 or 0 according as n = 0 or not. The d summation above is
1 CO
c °° f P - 2ninx fJv c
y y ! ™ = y
= i ;= _. J \c(x+j) + k + icy\2° ^
0
CO
C
J
_ CO
o - 2-n.inyv yVi.
_ c-2syl-2s | V o2ninklc
and the sum over k is c or 0 according as c\n or not. Moreover
f dv
J (V2 + ]
r(s) '

2.18 THE FUNCTIONAL EQUATION 39
and
00
r(s)
in the usual notation of Bessel functionsf.
We now have
B(z,s) = <t>(s)ys + i}/(s)y1-s + B0(z,s) (a > 1), B.18.4)
where re -i)
<j>(s) = 2CBs), rj,(s) = In * -?—¦? CBs - 1)
j r(s)
and
n^a^^in) cosBnnx) s"j; wyiy;. B.18.5)
n = i rw
We observe at this point that
for fixed a, whence the series B.18.5) is convergent for all s, and so
defines an entire function. Moreover we have
B0(z,s)<e-y (y^oo) B.18.6)
for fixed s. Similarly one finds
^.<e-y (y-co). B.18.7)
dy
We proceed to derive the 'Maass-Selberg' formula. Let D = {ze Jf:
\z\ ^ 1, \R(z)\ ^ ^} be the standard fundamental region for the modular
group, and let DY= {zeD: I(z)< Y), where Y> 1. Let R(s), R(a;) > 1
and write, for convenience, F = B(z, s), G = B(z, w). Then, according to
B.18.3), we have
Dy Dy
= \U
(FV2G-GV2F)dxdy
= (FVG-GVF)dn,
dDv
t see Watson, Theory of Bessel functions §6.16.

40 ANALYTIC CHARACTER OF C(s) AND Chap. II
by Green's Theorem. The integrals along x = + \ cancel, since
F(z + l) = F(z), G(z + 1) = G(z) (see B.18.1)). Similarly the integral for
\z\ = 1 vanishes, since F(-l/z) = F(z), G(-l/z) = G(z). Thus
B.18.8)
The functions ;ys andy1rs also satisfy the eigenfunction equation B.18.3)
(by B.18.2) with c = 0, d = 1) and thus, by B.18.4) so too does B0(z, s).
Consequently, if Z ^ Y, an argument analogous to that above yields
z *
where Fo = B0(z, s), Go = B0(z, w). Here we have used F0(z + 1) = F0(z)
and G0(z+ 1) = G0(z). (Note that we no longer have the corresponding
relations involving — l/z.) We may now take Z -* oo, using B.18.6) and
B.18.7), so that the first integral on the right above vanishes. On adding
the result to B.18.8) we obtain the Maass-Selberg formula
-s)-w(l-w)] \[b(z, s) B(z, w) dxdy
y2
= (s - w) }
B.18.9)

2.19 THE FUNCTIONAL EQUATION 41
where
2.19. In the general case there are now various ways in which one can
proceed in order to get the analytic continuation of <f) and \j/. However
one point is immediate: once the analytic continuation has been
established one may take w = 1 — s in B.18.9) to obtain the relation
- s) = iA(s)«A(l-s), B.19.1)
which can be thought of as a weak form of the functional equation.
The analysis we shall give takes advantage of certain special
properties not available in the general case. We shall take Y= 1 in
B.18.9) and expand the integral on the left to obtain
(s - w)<x(s + w)\j/(s)\J/(w) + jS(s, w)\{/(s) + y(s, w)\{/(w) + 5(s, w) = 0,
B.19.2)
where
<x(u) = (l-u) \\y-»dxdy-l= -2 \ A - x2)^1' u) dx
Dx 0
and jS, y, 3 involve the functions </> and BQ, but not \J/. If we know that ((s)
has a continuation to the half plane R(s) > <70 then <f)(s) has a
continuation to R(s) > ?<r0, so that a, jS, y, 5 are meromorphic there. If
(s - w)oc(s + w)\j/(w) + jS(s, w) = 0 B.19.3)
identically for R(s), R(w) > 1, then
«??> B.19.4)
(s - w)a(s + w)
which gives the analytic continuation oi\l/(w) to R(w) > ^<70. Note that
(s — w)tx(s + w) does not vanish identically. If B.19.3) does not hold for all
s and w then B.19.2) yields
_ _
which gives the analytic continuation of \p(s) to R(s) > \aQ, on choosing
a suitable w in the region R(u;) > 1. In either case C(s) may be continued
to R(s)>ffo-1. This process shows that C(s) has a meromorphic
continuation to the whole complex plane.
(s — w)(x(s + w)y/(w) + p(s, w)

42 ANALYTIC CHARACTER OF ?(s) AND Chap. II
Some information on possible poles comes from taking w = s in
B.18.9), so that B(z, w) = B(z, s). Then
+ Ba-l) .
If t ± 0 we may choose Y ^ 1 so that the second term on the right
vanishes. It follows that
for a ^\. Thus \jt is regular for a ^ \ and t ± 0, providing that </> is.
Hence C(s) has no poles for R(s) > 0, except possibly on the real axis.
If we take \ < R(s),R(u;)< 1 in B.19.5), so that <j)(s) and <f>(w) are
regular, we see that \j/(s) can only have a pole at a point s0 for which the
denominator vanishes identically in w. For such an s0, B.19.4) must
hold. However oc(u) is clearly non-zero for real u, whence ij/(w) can have
at most a single, simple pole for real w > ?, and this is at w = s0. Since it
is clear that ?(s) does in fact have a singularity at s = 1 we see that s0
= 1.
Much of the inelegance of the above analysis arises from the fact that,
in the general case where one uses the Eisenstein series rather than the
Epstein zeta-function, one has a single function p(s) = \j/(s)/4>(s) rather
than two separate ones. Here p(s) will indeed have poles to the left of
R(s) = \. In our special case we can extract the functional equation for
C(s) itself, rather than the weaker relation p(s)p(l — s) = 1 (see B.19.1)),
by using B.18.4) and B.18.5). We observe that
and that Ku{z) = K_u(z), whence n-sr(s)B0(z, s) is invariant under the
transformation s -»• 1 — s. It follows that
n-sr(s)B(z, s)-ns~ ^A -s)B{z, 1-s)
. -s
where we have written temporarily A(s) = 2n-sr(s)?Bs). The left-hand
side is invariant under the transformation z -> — 1/z, by B.18.1), and so,
taking z = iy for example, we see that A(s) = A(\ —s) and A(s —j) =
A(l —s). These produce the functional equation in the form B.6.4) and

2.20 THE FUNCTIONAL EQUATION 43
indeed yield
n-sr(s)B(z, s) = 7ts- 1r(l-s)B(z, 1 -s).
2.20. An insight into the nature of the zeta-function and its func-
functional equation may be obtained from the work of Tate [1 ]. He considers
an algebraic number field k and a general zeta-function
C(/,c)= [f{a)c{a)d*a,
where the integral on the right is over the ideles «7of k. Here /is one of a
certain class of functions and c is any quasi-character of J, (that is to
say, a continuous homomorphism from J to C x) which is trivial on k x.
We may write c(a) in the form co(a)\a\s, where cQ(a) is a character on J
(i.e. \co(a)\ = 1 for aeJ). Then co(a) corresponds to x> a 'Hecke
character' for k, and ?( /, c) differs from
-s\-l
(where P runs over prime ideals of k), in only a finite number of factors.
In particular, if k = Q, then C( f, c) is essentially a Dirichlet L-series
L(s, x)- Thus these are essentially the only functions which can be
associated to the rational field in this manner.
Tate goes on to prove a Poisson summation formula in this idelic
setting, and deduces the elegant functional equation
where f is the 'Fourier transform' of /, and c(a) = co(a) | a |: -s. The
functional equation for ?(s, x) may be extracted from this. In the case
k = Q we may take c0 identically equal to 1, and make a particular
choice f = f0, such that f0 = f0 and
The functional equation B.6.4) is then immediate. Moreover it is now
apparent that the factor n ~^sF(^s) should be viewed as the natural term
to be included in the Euler product, to correspond to the real valuation
of Q.
2.21. It is remarkable that the values of C(s) for s = 0, — 1, — 2,..., are
all rational, and this suggests the possibility of ap-adic analogue of C(s),
interpolating these numbers. In fact it can be shown that for any primep
and any integer n there is a unique meromorphic function Cp,n(s) defined

44 ANALYTIC CHARACTER OF ?(s) Chap. II
for selp, (the p-adic integers) such that
k^O, k = n(modp-l).
Indeed if n =jfe 1 (modp -1) then Cp,n(s) will be analytic on lp, and if n = 1
(modp — 1) then Cp,n(s) will be analytic apart from a simple pole at s = 1,
of residue 1 — A/p). These results are due to Leopoldt and Kubota [1 ].
While these p-adic zeta-functions seem to have little interest in the
simple case above, their generalizations to Dirichlet L-functions yield
important algebraic information about the corresponding cyclotomic
fields.

Ill
THE THEOREM OF HADAMARD AND
DE LA VALL3E POUSSIN, AND ITS
CONSEQUENCES
3.1. As we have already observed, it follows from the formula
that ?(s) has no zeros for ct > 1. For the purpose of prime-number
theory, and indeed to determine the general nature of ?(s), it is necessary
to extend as far as possible this zero-free region.
It was conjectured by Riemann that all the complex zeros of ?(s) lie
on the 'critical line' o = \. This conjecture, now known as the Riemann
hypothesis, has never been either proved or disproved.
The problem of the zero-free region appears to be a question of
extending the sphere of influence of the Euler product C.1.1) beyond
its actual region of convergence; for examples are known of functions
which are extremely like the zeta-function in their representation by
Dirichlet series, functional equation, and so on, but which have no
Euler product, and for which the analogue of the Riemann hypothesis
is false. In fact the deepest theorems on the distribution of the zeros
of ?(s) are obtained in the way suggested. But the problem of extending
the sphere of influence of C.1.1) to the left of ct = 1 in any effective
way appears to be of extreme difficulty.
00
71
where |/x(rz-) | < 1. Hence for ct near to 1
1
na
na ct—1
n=l
i.e. \Z(8)\
Hence if ?(s) has a zero on ct = 1 it must be a simple zero. But to prove
that there cannot be even simple zeros, a much more subtle argument
is required.
It was proved independently by Hadamard and de la Valle"e Poussin
in 1896 that ?(s) has no zeros on the line ct = 1. Their methods are similar
in principle, and they form the main topic of this chapter.

46 THE THEOREM OF HADAMARD Chap. Ill
The main object of both these mathematicians was to prove the
prime-number theorem, that as x -> oo
tt{x)
log re
This had previously been conjectured on empirical grounds. It was
shown by arguments depending on the theory of functions of a complex
variable that the prime-number theorem is a consequence of the
Hadamard-de la Vallee Poussin theorem. The proof of the prime -
number theorem so obtained was therefore not elementary.
An elementary proof of the prime-number theorem, i.e. a proof not
depending on the theory of ?(s) and complex function theory, has
recently been obtained by A. Selberg and Erdos. Since the prime-
number theorem implies the Hadamard-de la Vallee Poussin theorem,
this leads to a new proof of the latter. However, the Selberg-Erdos
method does not lead to such good estimations as the Hadamard-
de la Vallee Poussin method, so that the latter is still of great interest.
3.2. Hadamard's argument is, roughly, as follows. We have for a > 1
00
1 1
= y y —— = y -4-/E), C.2. i)
Z, Z, mpms Z-,ps^JK h K '
p m=l r v
where/(s) is regular for a > \. Since ?(s) has a simple pole at 5 = 1,
it follows in particular that, as a -»-1 (a > 1),
<3-2-2'
Suppose now that 5 = l+it0 is a zero of ?(s). Then if 5 = a-\-itQ, as
cr->l (cr> 1)
2 C°S{t°l?eP) = tog|«.)|-R/W ~ log(o-l). C.2.3)
p *
Comparing C.2.2) and C.2.3), we see that cos(?0log#) must, in some
sense, be approximately —1 for most values of p. But then cosBt0logp)
is approximately 1 for most values of p, and
log|C(a+2i«0)| ~
g
p p
so that l-\-2it0 is a pole of ?(s). Since this is false, it follows that
To put the argument in a rigorous form, let
1 p ycos(*0logp) n ycosB^0logj?)
^ p
p * p * p

3.2 AND DE LA VALLEE POUSSIN 47
Let 8', P'', Q' be the parts of these sums for which
Bk+lOT-a < t0logp < BJc+lOr+oc
for any integer k, and a fixed, 0 < a < \tt. Let S", etc., be the re-
remainders. Let A = 8'18.
If e is any positive number, it follows from C.2.2) and C.2.3) that
P< -(l-e)S
if a— 1 is small enough. But
P' > -8' = -X8
and P" > —^"cosa = — A— A)?cosa.
Hence __{A4-(l-A)cosa}ASf < -(l-e)8,
i.e. A— A)(l — cos a) < e.
Hence A -> 1 as a -» 1.
Also Q'^S'cos2<x, Q"^-8\
so that Q ^ S{\ cos 2a— 1+A).
Since A -> 1, 8 -> 00, it follows that Q -> 00 as <r -> 1. Hence \-{-2it0 is
a pole, and the result follows as before.
The following form of the argument was suggested by Dr. F. V.
Atkinson. We have
I xr^ cos(*0logj9)l2 = I y cos(^0logj3) 1 I2
2
P * P
p p
i.e. P*<
Suppose now that, for some t0, P'^log(a—1). Since S^->\og{l/{a~ 1)},
it follows that, for a given e and a—1 small enough,
(l-eJl0g2-L- < I((l + e)log-J- )i
\ 1 + 6 j cr—1
Hence Q -> 00, and this involves a contradiction as before.

48 THE THEOREM OF HADAMARD Chap. Ill
3.3. In de la Vallee Poussin's argument a relation between {,{o-\-it)
and t,{a-{-2it) is also fundamental; but the result is now deduced from
the fact that
3+4cos<?+cos2<? = 2(l+cos<?J > 0 C.3.1)
for all values of <f>.
w 1
We have Us) = exp "V 'V >
v Z* Z, mp™
00
and hence KE)l = exp ^ ^y
Hence
mp
p Tn = 1
ma
- expf V V 3+4cos(mt\ogp) + cosBmtlogp)\ „ 3
V\Z, Z, mpma ' ' ' '
v p m=l r '
Since every term in the last sum is positive or zero, it follows that
t?{o)\Z{o+it)\*\Z{o+2it)\ ^ 1 (a > 1). C.3.3)
Now, keeping t fixed, let a -> 1. Then
and, if 1+^ is a zero of ^(s), ?{o+it) = O(a-l). Also ?(<r+2#) = 0A),
since ^(s) is regular at l + 2i7. Hence the left-hand side of C.3.3) is
O(cr— 1), giving a contradiction. This proves the theorem.
There are other inequalities of the same type as C.3.1), which can
be used for the same purpose; e.g. from
5 + 8cos<? + 4cos2<?+cos3<? = (l + cos<?)(l + 2cos<?J > 0 C.3.4)
we deduce that
P(*)\Z(*+ft)\*\Uo+2ti)\*\Z(o+3it)\ > 1. C.3.5)
This, however, has no particular advantage over C.3.3).
3.4. Another alternative proof has been given by Ingham.f This
depends on the identity
)Z{3ai) W(n)\2 > ^4 ^
n8
where a is any real number other than zero, and
f Ingham C).

3.4 AND DE LA VALLEE POUSSIN 49
This is the particular case of A.3.3) obtained by putting ai for a and
—ai for 6.
Let ct0 be the abscissa of convergence of the series C.4.1). Then
ct0 ^ 1, and C.4.1) is valid by analytic continuation for a > cr0, the
function f(s) on the left-hand side being of necessity regular in this
half-plane. Also, since all the coefficients in the Dirichlet series are
positive, the real point of the line of convergence, viz. 5 = cr0, is a
singularity of the function.
Suppose now that l-{-ai is a zero of ?(s). Then 1—ai is also a zero,
and these two zeros cancel the double pole of ?2(s) at 5 = 1. Hence/(s)
is regular on the real axis as far as 5 = — 1, where ?Bs) = 0; and so
ct0 = —1. This is easily seen in various ways to be impossible; for
example C.4.1) would then give/(?) ^ 1, whereas in fact/(?) = 0.
3.5. In the following sections we extend as far as we can the ideas
suggested by § 3.1.
Since ?(s) has a finite number of zeros in the rectangle 0 < a ^ 1,
0 ^ t < T and none of them lie on a = 1, it follows that there is a
rectangle 1—S^ct^I, 0 ^ t ^ T, which is free from zeros. Here
S = B(T) may, for all we can prove, tend to zero as T -> oo; but we can
obtain a positive lower bound for B(T) for each value of T.
Again, since l/?(s) is regular for o = 1, 1 ^ t ^ T, it has an upper
bound in the interval, which is a function of T. We also investigate
the behaviour of this upper bound as t -> oo. There is, of course, a similar
problem for ?(s), in which the distribution of the zeros is not imme-
immediately involved. It is convenient to consider all these problems
together, and we begin with ?(s).
Theorem 3.5. We have
?(«) = O(log*) C.5.1)
uniformly in the region
where A is any positive constant. In particular
t(l+it) = O{logt). C.5.2)
In B.1.3), take a > 1, a = N, and make 6 ->oo. We obtain
^ 1 Fl 1— 4-1 N1-8
?-4 ns J x3*1 s—1

50 THE THEOREM OF HADAMARD Chap. Ill
the result holding by analytic continuation for ct > 0. Hence for ct > 0,
* > 1> „
In the region considered, if n ^ t,
\n~s\ = n-° = e-aloen < expj — A —-^-Jlogn] < n~xeA.
Hence, taking N = [t],
= O(logN)+O(l) = O(\ogt).
This result will be improved later (Theorems 5.16, 6.11), but at the
cost of far more difficult proofs.
It is also easy to see that
V(8) = O(log20 C.5.5)
in the above region. For, differentiating C.5.3),
N
and a similar argument holds, with an extra factor log t on the right-hand
side. Similarly for higher derivatives of ?(s).
We may note in passing that C.5.3) shows the behaviour of the
Dirichlet series A.1.1) for a < 1. If we take a = 1, t =fc 0, we obtain
1 N
J
N
which oscillates finitely as N -> oo. For a < 1 the series, of course,
diverges (oscillates infinitely).
3.6. Inequalities for l/?(s), ?'(«)/?(«), and log ^E)- Inequalities of
this type in the neighbourhood of ct = 1 can now be obtained by a slight
elaboration of the argument of § 3.3. We have for ct > 1
1 < g()}*|?(+2«)|i 0(i^U C.6.1)

3.6 AND DE LA VALLEE POUSSIN 51
Also ?{l+it)-?(a+it) = - J t,\u+it) du = O{(a~~l)\ogH} C.6.2)
for a > 1— A/log t. Hence
The two terms on the right are of the same order if a—I = log-9*.
Hence, taking cr—1 = ^log"9*, where A3 is sufficiently small,
|?A+it)\ > A log-7*. C.6.3)
Next C.6.2) and C.6.3) together give, for 1—,4 log* < a < 1,
|?(cr+i*)| > A\og-7t—A{l—a)logH, C.6.4)
and the right-hand side is positive if I—a < J.log~9*. Hence ?(s) has
no zeros in the region a > 1—.4 log"9*, and in fact, by C.6.4),
-L = O(log7*) C.6.5)
in this region.
Hence also, by C.5.5),
= O(log9*), C.6.6)
and log C(j) = f ^±^} dt*+log CB+*<) = O(log90, C.6.7)
2
both for a > 1— ,4 log-9*.
We shall see later that all these results can be improved, but they
are sufficient for some purposes.
3.7. The Prime-number Theorem. Let tt{x) denote the number of
primes not exceeding x. Then as x -> oo
t^^JL.. C.7.1)
log re
The investigation of 7t{x) was, of course, the original purpose for
which ?(s) was studied. It is not our purpose to pursue this side of the
theory farther than is necessary, but it is convenient to insert here a
proof of the main theorem on tt{x).
We have proved in A.1.3) that, if a > 1,
2
We want an explicit formula for tt(x), i.e. we want to invert the above

52 THE THEOREM OF HADAMARD Chap. Ill
integral formula. We can reduce this to a case of Mellin's inversion
formula as follows. Let
Then i»iiW_CBW = J^(fa. C.7.2)
This is of the Mellin form, and u>{s) is a comparatively trivial function;
in fact since tt(x) ^ x the integral for to(s) converges uniformly for
a ^ \Jr'&, by comparison with
00
f dx
Hence u>(s) is regular and bounded for a ^ ?-{-S. Similarly so is tt/(s),
since m
,'(«) = J n(x)logxJl{J*iJdz.
U) (S) =
2
We could now use Mellin's inversion formula, but the resulting
formula is not easily manageable. We therefore modify C.7.2) as
follows. Differentiating with respect to 5,
_Hi) ipg|(?) ,
w J x
Denote the left-hand side by </>(s), and let
J xs+1
2
g(x) = ["W°S*au, h(x) = [?Mdu,
J u J u
0 0
tt{x), g{x), and h(x) being zero for x < 2. Then, integrating by parts,
00 00
«?E) = J sr'(a;)a;-s cZrc = 5 J g(x)x~a-1 dx
o
00 00
= s ( h'{x)x-a dx = s2 J ^(a:)^-8-1 cZrc (a > 1),
o
00
1\ /* Xi / \
""Oj I /&(•*/) » 1 T
2- I " ¦ ¦¦ " JU KajJU *
5 1

3.7 AND DE LA VALLEE POUSSIN 53
Now h{x) is continuous and of bounded variation in any finite interval;
and, since 7r(x) < re, it follows that, for re > 1, gr(rc) < re log re, and
h{x) ^ re log re. Hence ^(rejre*-2 is absolutely integrable over @,oo) if
k < 0. Hence
Jfc + too
h(x) 1 f 6A—s) . , „
x 2tti J A—sJ
A;—ioo
or h{x) = —. I
2ttI J
c—ioo
The integral on the right is absolutely convergent, since by C.6.6) and
C.6.7) <f){s) is bounded for a ^ 1, except in the neighbourhood of 5 = 1.
In the neighbourhood of 5 = 1
and we may write </>(s) =
5— 1
where «/»(s) is bounded for cr^l, \s—1|>1, and ^(s) has a logarithmic
infinity as 5 -> 1. Now
c+ioo c+ico
; 2irt J (s—1)«2 ^27rt J
i i
J () J
c—ioo c—ioo
The first term is equal to the sum of the residues on the left of the
line R(s) = c, and so is
re—log re—1.
In the other term we may put c = 1, i.e. apply Cauchy's theorem to
the rectangle (ldz^, c^iT), with an indentation of radius e round
s = 1, and make T -> oo, e -> 0. Hence
00
The last integral tends to zero as re ~> oo, by the extension to Fourier
integrals of the Biemann-Lebesgue theorem.f Hence
h(x)~x. C.7.3)
f See my Introduction to the Theory of Fourier Integrals, Theorem 1.

54 THE THEOREM OF HADAMARD Chap. Hi
To get back to tt{x) we now use the following lemma:
Let f(x) be be positive non-decreasing, and as x -> oo let
^ re.
1
/(re) ~ re.
If 8 is a given positive number,
X
A-8)* < J®^< A+3)* (*>*0(8))
Hence for any positive e
/* -Pint \ /* -/v \ /* -^/ \
I ¦ ' ' KajiJU —^ I •¦——— tv 66 ^^ I • UulL
= BS+e+Se)rc.
But, since/(re) is non-decreasing,
Hence /(re) < rc(l+e)/l+8 + — \.
Taking, for example, e = VS, it follows that
re
X
Similarly, by considering f-^
@
we obtain lim^i-^ > 1,
— x
and the lemma follows.
Applying the lemma twice, we deduce from C.7.3) that
g(x) ~ x,
and hence that 7r(rc)logrc ^ x.
3.8. Theorem 3.8. There is a constant A such that ?(s) is not zero for

3.8 AND DE LA VALLEE POUSSIN 55
We have for a > 1
Hence, for a > 1 and any real y,
-~l?{3+4cos(mrlog^)+cosBmylog^)} > 0. C.8.2)
Now _*>><.* +0(i). C.8.3)
4(a; a—1
Also, by B.12.7),
fg fe)
where p — j8+*y runs through complex zeros of ?(«). Hence
((
Since every term in the last sum is positive, it follows that
C'8-5)
and also, if j8+iy is a particular zero of ?(s), that
From C.8.2), C.8.3), C.8.5), C.8.6) we obtain
a—1 a—p
or say > —^^ogy.
a—1 <7—p
Solving for @, we obtain
1 — 6 >
The right-hand side is positive if a—1 = ^1/logy, and then
logy
the required result.

56
THE THEOREM OF HADAMARD
Chap. Ill
3.9. There is an alternative method, due to Landau,f of obtaining
results of this kind, in which the analytic character of ?(s) for a ^ 0
need not be known. It depends on the following lemmas.
Lemma a. If f(s) is regular, and
<eM (M > 1)
in the circle \s—so| ^ r, then
s-p
AM
\s~so
where p runs through the zeros off(s) such that \p—so\ < \r.
The function g(s) = f(s) JJ {s—p)-1 is regular for |s—so| < r, and not
p
zero for \s—so| ^ \r. On \s—so| = r, |5—p\ ^ \r ^ |s0—p\, so that
f{s) T-r (So-p\
f(so)l i\s-p)
<
This inequality therefore holds inside the circle also. Hence the function
*(,) > log(Mj,
where the logarithm is zero at 5 = s0, is regular for \s—so\ ^ \r, and
h{s0) = 0, R{h{s)} < M.
Hence by the Borel-Caratheodory theorem^
\h(s)\ < AM (\s-so\ < $r), C.9.1)
and so, for \s—so\ < \r,
\h'{s)\ =
- f
2ni J
Hz)
dz
AM
2ni J B—sJ
\z—s\ = \r
This gives the result stated.
Lemma j8. // f(s) satisfies the conditions of the previous lemma, and
has no zeros in the right-hand half of the circle \s—so\ < r, then
nlf'(80)\ AM
while if f{s) has a zero pQ between so—\r and s0, then
-R
\f'{so)\ ^ AM 1
t Landau A4).
f(s0) i r So—Po
X Titchmarsh, Theory of Functions, § 5.5.

3.9 AND DE LA VALLEE POUSSIN
Lemma a gives
I/(So) I r ^ so-p
and since R{l/(s0—p)} > 0 for every p, both results follow at once.
Lemma y. Let f(s) satisfy the conditions of Lemma a, and let
57
/'(So)
M
Suppose also thatf{s) =?¦ 0 in the part a ^ a0— 2r' of the circle \s—so|
where 0 < r' < \r. Then
f'{s) .M
/(s) r
Lemma a now gives
r,
for all 5 in \s—so\ ^ ^r, a ^ a0—2r', each term of the sum being
positive in this region. The result then follows on applying the Borel-
Caratheodory theorem to the function —f'{s)jf{s) and the circles
\s—so\ = 2r', \s-so\ = r'.
3.10. We can now prove the following general theorem, which we
shall apply later with special forms of the functions 6(t) and <f>(t).
Theorem 3.10. Let
as t-^- co in the region
I —6{t) < a < 2 (« > 0),
where cf>(t) and l/B(t) are positive non-decreasing functions of t for t
such that 6(t) ^ 1, <f>(t) -> 00,
0,
iAere *5 a constant Ax such that l,(s) has no zeros in the region
Let fi+iy be a zero of ?(s) in the upper half-plane. Let
() ^ 2,
r = BBy+1).

58
THE THEOREM OF HADAMARD
Chap. Ill
Then the circles \s—so\ < r, \s—sQ\ < r both lie in the region
a > l—6(t).
Now
"o-l
<
and similarly for s'o. Hence there is a constant A2 such that
as)
in the circles \s—sQ\ < r, \s—sQ\ < r respectively. We can therefore
apply Lemma /? with M = A2<f>By-{-l). We obtain
and, if
C.10.3)
C.10.4)
C.10.5)
Also as cr0 -> 1
Hence
ct0—1
No-
NoC.10.6)
where a can be made as near 1 as we please by choice of a0.
Now C.8.2), C.10.3), C.10.5), and C.10.6) give
3a . 5A«<t>By4-l) 4 . n
<70—
*o-jS
3a
[4(ao-l) ' 4
^y+l)!
,-1) ' 4 0By+l)j
3a
3a
To make the numerator positive, take a = |, and
1 0By+l)
ao-l =
this being consistent with the previous conditions, by C.10.1), if y is
large enough. It follows that

3.10 AND DE LA VALLEE POUSSIN 59
as required. If C.10.4) is not satisfied,
0 5
which also leads to C.10.2). This proves the theorem.
In particular, we can take 6(t) = \, <f>(t) = log(i+2). This gives a
new proof of Theorem 3.8.
3.11. Theorem 3.11. Under the hypotheses of Theorem 3.10 we have
In particular
C.11.1), C.11.2)
uniformly for °>^
C.11.4), C.11.5)
We apply Lemma y, with
In the circle \s—so\ < r
and „, °. — 01-
U-1J \eBto+S)j--U[ r
We can therefore take M = A(j)Bt0 + 3). Also, by the previous theorem,
C(s) has no zeros for
Kfn a>1 A e{2(to+l)+l} _
« < tQ+l, a ^ l-AXJ(—-—-- - I-Ay
Hence we can take 2r' =
Hence ~-r =
for \s—sQ\ :
and in particular for
c = tQ, a ^ 1 —
4
This is C.11.1), with t0 instead of t.

60 THE THEOREM OF HADAMARD Chap. Ill
Abo, if 1-^
4
1 +
f
J
1
A<f>Bl+3)
Hence C.11.2) follows if a is in the range C.11.6); and for larger a it
is trivial.
Since we may take 6(t) == I, <f>(t) = log(<+2), it follows that
| OtlO C.11.7), C.11.8)
in a region a ^ 1—^4/logi; and in particular
C.11.9), C.11.10)
3.12. For the next theorem we require the following lemma.
00
Lemma 3.12. Let f{s) =
s
where an = 0{ijj{n)}, ip(n) being non-decreasing, and
of
na l(cr— lW
n=l v ' '
as or-> 1. TAen if c > 0, cr+c > 1, # is noi an integer, and N is the
integer nearest to x,
c+iT
ns 2ni J
n J w
n<Cx c — iT

3.12 AND DE LA VALLEE POUSSIN 61
// x is an integer, the corresponding result is
If* ill / j*» ^
—-j — = —: I f(s-\-w)—dw4-0{-= > 4-
• ns 2x? 2m J w \T{a-\-c—l)aJ
71=1 c-iT
Suppose first that x is not an integer. If n < x, the calculus of residues
C — iT C + iT — oo + iT.
-M f + f + f
-oo-iT c-iT c + iT
Now
c+iT . c+iT
I I — I I i —I —— I I — I
111 II I \ I 1 / 111 O
j \aj w LlvluBx/aJ -aa+iT 1{Jgx/n j \aj w
(x/ny 1,-1 (z/n)' f du,
In) I log x 7i J
= 0
and similarly for the integral over (—co—iT,c—iT). Hence
c + iT
if \ ^^ 1 /~\ I
J7ri J \n) w I
c + iT
2-ni
c-iT
If 7i > x we argue similarly with —oo replaced by +°o, and there is
no residue term. We therefore obtain a similar result without the term 1.
Multiplying by aa n~s and summing,
c+iT
1 f
—.
2tu J
00
— dw = > —4-01 —
n<x
If n < ^a: or n > 2#, j log a;/rz. j > ^4, and these parts of the sum are
ol
g J^i) = o(^^
If lY < n ^ 2a;, let 7i = N+r. Then
iV+r ^4r Ar

62 THE THEOREM OF HADAMARD Chap. Ill
Hence this part of the sum is
A similar argument applies to the terms with \x < n < N. Finally
aN\ _Q( iMl\) ) ^itLlNW-o-c)
Na+C\logx/N\ \Na+clog{l + (x—N)IN}} \ \x—N\
Hence C.12.1) follows.
If x is an integer, all goes as before except for the term
c + iT
Zrrix? ) w 2-niv?
i
c—iT 27Tixs\ \Tli
c-iT
Hence C.12.2) follows.
3.13. Theorem 3.13. We have
00 .
Z(S) ?, n3
at all points of the line a = 1.
Take an = fji{n), a = 1, a = 1, in the lemma, and let x be half an odd
integer. We obtain
c + iT
sr* fj.{n) 1 T 1 xw j nf xC\,s)fl°gx\
n<x c_iT
The theorem of residues gives
c + iT ,-8-iT ~8 + iT c+iT,
ir i xw, i i/f,r,f\
I din I I I -I— I -1— I I
2ni J ?,(s-\-w) w ?(s) 2-ni \ J J J I
c-iT ^c~iT ~8-iT -t'-™'
if 8 is so small that ?,(s-\-w) has no zeros for
R{w) > —8, |IE+w?)| < \t\ + T.
By § 3.6 we can take 8 == A log~9T. .Then
~8+iT , T
r i xw I C
j ciw = O\x~°log7T j
J t,(s-^-w) w \ J
T/8
J J{l+v2)
-TI8

3.13 AND DE LA VALLEE POUSSIN 63
and
c+iT
c .
J ?(s+w) w \ T J I \ T )'
+iT x 8 '
J ?(+) \ J
-8+iT x -8
and similarly for the other integral. Hence
Take c = 1/logz, so that x° = e; and take T = exp{(log a;I'10}, so that
log T = (log*I/10, 5 = A(logx)-*l10, Xs = T^. Then the right-hand side
tends to zero, and the result follows.
In particular y ^—- = 0.
, n
3.14. The series for ?'(s)/?(s) and log ?(s) on a = 1.
Takingf an =A{n) = O(logn), a = l,a = 1, in the lemma, we obtain
c + iT
f
J
J
In this case there is a pole at w = 1—5, giving a residue term
where a is a constant. Hence if 5 =?¦ 1 we obtain
Taking c = I/log #, T = exp{(log xI/10}, we obtain as before
ns Us) 1—5
The term x1-s/{l—s) oscillates finitely, so that if R(s) = 1, 5 ^= 1, the
series ^Afojn'3 is not convergent, but its partial sums are bounded.
If 5 = 1, we obtain
g() C.14.2)
t n
n<x
or, since
log]) r ^ logp _ ^ log^
+Z Z^Z"
V l^KE = logx+O(l). C.14.3)
Z p
t See A.1.8).

64 THE THEOREM OF HADAMARD Chap. Ill
Since A1(n) = A(n)/\ogn, and I/logn tends steadily to zero, it follows
that . , x
is convergent on a = 1, except for t = 0. Hence, by the continuity
theorem for Dirichlet series, the equation
log ?(8) =
n=2
holds for a = 1, t ^ 0.
To determine the behaviour of this series for 5 = 1 we have, as in the
case of l/?(s),
c + iT
*—/ 71 JiTTl J W \ 1 I
where c = I/log x, and T is chosen as before. Now
c+iT ,-8-iT -8 + iT c + iT.
if xw ilc c r\if
—- log ?{w+l) —dw = ¦—[ 4. -f -j. I
2tt* J 2^ 27ri\ J J J I 2tti J
where C is a loop starting and finishing at 5 = —8, and encircling the
origin in the positive direction. Defining 8 as before, the integral along
a = —8 is O(x~8log10T), and the integrals along the horizontal sides
are O(^T~1log9 T), by C.6.7). Since
is regular at the origin, the last term is equal to
2-ni
c
Since
1 r . 1 xw ,
—: log dW.
7Tl J W W
c
If, 1 dw 1 a i 9
— log = . Ac log2w
TTl J W W 4lVI
c
= L{log2(Se*v)—log2(Se-^)} = —logS,
this term is also equal to
I r 1 xw—l
—: I log dw—log8.
2m J w w
c
Take C to be a circle with centre w = 0 and radius p (p < 8), together

3.14 AND DE LA VALLEE POUSSIN 65
with the segment (—8,—p) of the real axis described twice. The
integrals along the real segments together give
f () \
* f iog(-V) ^ *,_ _L \ log(
m J &\ue-™] ~u 2tti J &\ue
8
f iog(V) *, \ log( du
2m J &\ue-™] ~u 2tti J &\ue™) —u
8 P
8 8 log x
g
= — du = — I
J u J v
p p log x
X 8 log x
dv
g
— dv~ —
p log x X
= y+log(81og:r)+o(l)
if p log x -> 0 and 8 log x -> oo. Also
log dw = 01 plog-log x).
J w w \ P I
\w\"p
Taking p = l/log2:r, say, it follows that
^ = loglog.T+y+o(l). C.14.4)
n<x
The left-hand side can also be written in the form
yi+y y _L.
p<x -1 ot>2 pm<x
As x ->¦ oo, the second term clearly tends to the limit
2
771 = 2 P
iAV41vv , — = loglog:r-|-y— y y \-o(l). C.14.5)
^-i p
p<x* m=2 p
3.15. Euler's product on o- = 1. The above analysis shows that
for a = 1, t =? 0,
qs
p * <i 1
where p runs through primes and q through powers of primes. In fact
the second series is absolutely convergent on a = 1, since it is merely
a rearrangement of « ,
V V
mpms
p.m = 2. r

66 THE THEOREM OF HADAMARD Chap. Ill
which is absolutely convergent by comparison with
p m = 2 p
Hence also
ps jL* Z-, nip™
P p
mp
ms
P
Taking exponentials,
i.e. Euler's product holds on a = 1, except at t = 0.
At s = 1 the product is, of course, not convergent, but we can obtain
an asymptotic formula for its partial products, viz.
PK)
log a;'
To prove this, we have to prove that
1\
C.15.2)
f{x) = —log
Now we have proved that
0(a) = y;-^ = loglog^+y+o(l).
Also
\ y ±+l
2
y
mp'n'
P m~2
pm>x
which tends to zero as x -> oo, since the double series is absolutely
convergent. This proves C.15.2).

3.15 AND DE LA VALLEE POUSSIN 67
It will also be useful later to note that
C.15.3)
¦NX ¦ Pi ™*
For the left-hand side is
11/p \ ^j
Note also that C.14.3), C.14.5) with error term 0A), and C.15.2)
can be proved in an elementary way, i.e. without the theory of the
Riemann zeta-function; see Hardy and Wright, The Theory of Numbers
Eth edn), Theorems 425 and 427-429. Indeed the proof of Theorem 427
yields C.14.5) with the error term 0\ )
\logxj'
NOTES FOR CHAPTER 3
3.16. The original elementary proofs of the prime number theorem
may be found in Selberg [7] and Erdos [1], and a thorough survey of the
ideas involved is given by Diamond [1]. The sharpest error term
obtained by elementary methods to date is
n(x) = Li(x) + 0 [x exp{ - (log x)^E} ], C.16.1)
for any ? > 0, due to Lavrik and Sobirov [1]. Pintz [1] has obtained a
very precise relationship between zero-free regions of ?(s) and the error
term in the prime-number theorem. Specifically, if we define
R(x) = max{|rc@-Li@|: 2 ^ t ^ x},
then
min ((* ~0)lo%x + log |y|}, (x -» 00),
the minimum being over non-trivial zeros p of ?(s). Thus C.16.1) yields
for any p and any x. Now, on taking log x = A — /7) loglyl we deduce
for any s' > 0. This should be compared with Theorem 3.8.
3.17. It may be observed in the proof of Theorem 3.10 that the bound
C(s) = O(e^) is only required in the immediate vicinity of s0 and s'o. It
would be nice to eliminate consideration of s'o and so to have a result of

68 THE THEOREM OF HADAMARD Chap. Ill
the strength of Theorem 3.10, giving a zero-free region around 1 + it
solely in terms of an estimate for ?(s) in a neighbourhood of 1 + it.
Ingham's method in § 3.4 is of special interest because it avoids any
reference to the behaviour of ?(s) near l + 2iy. It is possible to get
quantitative zero-free regions in this way, by incorporating simple sieve
estimates (Balasubramanian and Ramachandra [1]). Thus, for exam-
example, the analysis of § 3.8 yields
? -^{1 + cos(mylogp)} ^ - + O(logy).
p.m. P G~l °-P
However one can show that
cos(ylogp)}>
X<p<2X
for X $s y2, by using a lower bound of Chebychev type for the number of
primes X <p ^ 2X, coupled with an upper bound O(h/\og h) for the
number of primes in certain short intervals X < p ^ X + h. One then
derives the estimate
^{l + cos(ylogp)}>
p>y p a -I
and an appropriate choice of a = 1 + (A /log y) leads to the lower bound
3.18. Another approach to zero-free regions via sieve methods has
been given by Motohashi [1]. This is distinctly complicated, but has the
advantage of applying to the wider regions discussed in § § 5.17, 6.15 and
6.19.
One may also obtain zero-free regions from a result of Montgomery
[1; Theorem 11.2] on the proliferation of zeros. Let n(t, w, h) denote
the number of zeros p = /? + iy of ?(s) in the rectangle 1 — w ^ /? ^ 1,
t — ^h < y ^ t + \h. Suppose p is any zero with fi > ?, y > 0, and that 5
satisfies 1 — A ^ 5 ^ (log y) ~ i. Then there is some r with 5 ^ r ^ 1 for
which 3
n(y, r, r) + nBy, r, r) > r C.18.1)
z(l p)
Roughly speaking, this says that if 1 - /? is small, there must be many
other zeros near either 1 + iy or l + 2iy. Montgomery gives a more
precise version of this principle, as do Ramachandra [1] and
Balasubramanian and Ramachandra [3], To obtain a zero-free region

3.18 AND DE LA VALLEE POUSSIN 69
one couples hypotheses of the type used in Theorem 3.10 with Jensen's
Theorem, to obtain an upper bound for n(t, r, r). For example, the bound
, T= 1*1 + 2,
which follows from Theorem 4.11, leads to
n (t, r,r)<?r log T + loglog T + log -. C.18.2)
r
On choosing S = (loglog y)/(log y), a comparison of C.18.1) and C.18.2)
produces Theorem 3.8 again.
One can also use the Epstein zeta-function of §2.18 and the
Maass-Selberg formula B.18.9) to prove the non-vanishing of ?(s) for
a = 1. For, if s = ? + it and cf)(s) = 2?Bs) = 0, then
by the functional equation B.19.1). Thus B.18.9) yields
JJ
D
for any w j= s, 1—s. This, of course, may be extended to w = s or
w = 1 — s by continuity. Taking w = \ — it = s we obtain
^odxdy n
D
so that B(z, s) must be identically zero. This however is impossible since
the Fourier coefficient for n = 1 is
according to B.18.5), and this does not vanish identically. The above
contradiction shows that ?Bs) =/= 0. One can get quantitative estimates
by such methods, but only rather weak ones. It seems that the proof
given here has its origins in unpublished work of Selberg.
3.19. Lemma 3.12 is a version of Perron's formula. It is sometimes
useful to have a form of this in which the error is bounded as x -> N.

70 THE THEOREM OF HADAMARD Chap. Ill
Lemma 3.19. Under the hypotheses of Lemma 3.12 one has
c + iT
f( )
K f f(s + w)
C — iT
This follows at once from Lemma 3.12 unless x — N = O(x/T). In the
latter case one merely estimates the contribution from the term n = N
as
C+lT C+iT
f aN( x\wdw _ [ aN( fw\\dw
— iT C —
and the result follows.

IV
APPROXIMATE FORMULAE
4.1. In this chapter we shall prove a number of approximate formulae
for ?(s) and for various sums related to it. We shall begin by proving
some general results on integrals and series of a certain type.
4.2. Lemma 4.2. Let F(x) be a real differentiable function such that
F'{x) is monotonic, and F'{x) > w > 0, or F'{x) < —m < 0, throughout
the interval [a, b ]. Then
b
dx
a
m
D.2.1)
Suppose, for example, that F'{x) is positive increasing. Then by the
second mean-value theorem
6 6
a
a
- J_ f F'
~ F'(a) J *
dx -
a
and the modulus of this does not exceed 2/ra. A similar argument
applies to the imaginary part, and the result follows.
4.3. More generally, we have
Lemma 4.3. Let F(x) and 0{x) be real functions, G{x)/F'(x) monotonic,
and F'{x)jG(x) > m > 0, or < — m < 0. Then
b |
J G{x)eiFW dx
a
m
The proof is similar to that of the previous lemma.
The values of the constants in these lemmas are usually not of any
importance.
4.4. Lemma 4.4. Let F{x) be a real function, twice differentiable, and
let F"{x) > r > 0, or F"{x) < — r < 0, throughout the interval [a, b].
Then b
f ei
dx
a
D.4.1)

72 APPROXIMATE FORMULAE Chap. IV
Consider, for example, the first alternative. Then F'{x) is steadily
increasing, and so vanishes at most once in the interval {a, b), say at c.
b c_s c+s b
1 = J e«w <fo; ^—^ J -i- J _!_ J ^— A
a « c-S c+8
where 8 is a positive number to be chosen later, and it is assumed that
a+8 < c < 6—8. In 73
F'{x) = j F"{t) dt > r{x—c) > rB.
4
Hence, by Lemma 4.2, |/3| <-x-
7*0
7j satisfies the same inequality, and |/2| < 28. Hence
Taking 8 = 2?—?, we obtain the result. Ifc < a-j-S, ore > b—8, the
argument is similar.
4.5. Lemma 4.5. Let F(x) satisfy the conditions of the previous lemma,
and let G{x)/F'{x) be monotonic, and \0{x)\ ^ M. Then
<8M_
J Q{x)eiF&> dx
a
The proof is similar to the previous one, but uses Lemma 4.3 instead
of Lemma 4.2.
4.6. Lemma 4.6. Let F(x) be real, with derivatives up to the third order.
Let 0 < A2 < F"{x) < AX2, D.6.1)
or 0 < A2 ^ —F"(x) < AX2, D.6.2)
and \F'"{x)\^AX3, D,6.3)
throughout the interval (a,b). Let F'(c) = 0, where
a < c < 6. D.6.4)
Then in the case D.6.1)
b
f , piiir+iF(c)
J <™ *5
a
^))- D6-5)
In the case D.6.2) Z/ie factor e\in is replaced by e~\in. If F'(x) does not
vanish on [a, b] then D.6.5) holds without the leading term.

4.6
APPROXIMATE FORMULAE
73
If F'(x) does not vanish on [a, b] the result follows from Lemmas 4.2
and 4.4. Otherwise either D.6.1) or D.6.2) shows that F'(x) is monotonic,
and so vanishes at only one point c. We put
6 c-S c+S 6
a
f j j
a c—8 c+8
assuming that a-\-8 < c ^ 6—5. By D.2.1)
c+S
c+S
c-8
SimUar.y
J = o{±).
a
Also
c+S c+S
f = f exV[i{F{c)+{x-c)F'{c)+\{x-cYF"{c)+
X 8
—X —
c+8
= eiF^ \
c-8
c+8
J e
c-8
Supposing F"{c) > 0, and putting
%{x-cfF"{c)
the integral becomes
= u,
^ ^ = {F(^l J ^
Taking S = (A2A3)-5, the result follows.
If 6—S<c^6, there is also an error
c+S
= O
and also
and similarly ifa

74 APPROXIMATE FORMULAE Chap. IV
4.7. We now turn to the consideration of exponential sums, i.e. sums
of the form
where f(n) is a real function. If the numbers f(n) are the values taken
by a function f(x) of a simple kind, we can approximate to such a sum
by an integral, or by a sum of integrals.
Lemma 4.7.f Let f(x) be a real function with a continuous and steadily
decreasing derivative f'(x) in (a,b), and let f'(b) = a, /'(a) = jS. Then
b
f e2^tf<*>-ra><Za;+0{log(j8-a+2)}, D.7.1)
77 is any positive constant less than 1.
We may suppose without loss of generality that 77 — 1 < a <C 77, so
that v > 0; for if k is the integer such that 77 —1 < a—k ^ 77, and
h(x) = f(x)—kx,
then D.7.1) is
V e2Trih(n) __ V
a<n*ib a'~ ij<v —
where a' = a—k, j8' = jS—k, i.e. the same formula for h(x).
In B.1.2), let <f>(x) = e2™™. Then
2
a<n<b
sin 2wx
Also x~\pc]~:k — —
if x is not an integer; and the series is boundedly convergent, so that
we may multiply by an integrable function and integrate term-by-term.
Hence the second term on the right is equal to
v-l
a
The integral may be written
6 6
1 f f'(x) 1 f f'i
_ J K ' Mc2ni(f(x)-vx)} i_ _J_1
a a
f van der Corput A).

4.7 APPROXIMATE FORMULAE 75
fix)
Since ; is steadily decreasing, the second term is
/ {x)~rv
o(J-\
by applying the second mean-value theorem to the real and imaginary
parts. Hence this term contributes
= 0{Iog(/3+2)} + 0(l).
Similarly the first term is 0{fij(v—f3)} for v > j3+ij, and this contributes
l y -e-) = o(
o
Finally
i e27ri{/(x)-,x}/'^ ^ = y « , y
"J ,4iL 2^, Ja z.j
and the integrated terms are O{log(j8+2)}. The result therefore follows.
4.8. As a particular case, we have
Lemma 4.8. Let f(x) be a real differentiate function in the interval [a, 6],
letf'{x) be monotonic, and let \f'{x)\ < 6 < 1. Then
b
= \ eZir*te) dx+O{\). D.8.1)
Taking 17 < 1—6, the sum on the right of D.7.1) either reduces to the
single term v = 0, or, if/\x) > 17 or ^ —17 throughout [a, 6], it is null,
and 6
J e2^/(x) fa = 0A)
a
by Lemma 4.2.
4.9. Theorem 4.9.f Le? /(x) 6e a real function with derivatives up to the
third order. Let f'(x) be steadily decreasing in a ^ x ^ b, and f'(b) = a,
f'(a) = /?. Let xv be defined by
I van der Corput B).

76 APPROXIMATE FORMULAE Chap. IV
Let A2 < \j"{x)\ < AX2, \f'"(x)\ < A\3.
Then
i{f(xv)-vxv}
+ O[log{2+F—a)X2}]+0{(b—a)Aj A*}.
We use Lemma 4.7, where now
j8_a = 0{F-a)A2}.
Also we can replace the limits of summation on the right-hand side by
(a-f I,j8—1), with error O(A2i). Lemma 4.6. then gives
b
^T e2ni{f(x)-vx) fa _ e-\iir V
<v<B-l { a+l<v<
a+KKjS-1
2 l
The second term on the right is
|A|
and the last term is
0{logB+j8--a)} = O[log{2 + F-a)A2}].
Finally we can replace the limits (<x-\-l,fi—l)hy (a,j6] with error
4.10. Lemma 4.10. Let f(x) satisfy the same conditions as in Lemma 4.7,
and let g(x) be a real positive decreasing function, with a continuous
derivative g'(x), and let \g'(x)\ be steadily decreasing. Then
b
f g(
+ O{g(a)log(P- cc+2)}+ O{\ g'(a) |}.
We proceed as in § 4.7, but with
We encounter terms of the form
b
I U 1 *A*> t i
J J V
a
6
and also

4.10 APPROXIMATE FORMULAE 77
The former lead to O{g(a)log(^—a-f 2)} as before. The latter give, for
example,
and the result follows.
4.11. We now come to the simplest theoremf on the approximation
to ?(s) in the critical strip by a partial sum of its Dirichlet series.
Theorem 4.11. We have
uniformly for a ^ cr0 > 0, \t\ < 2ttxJC, when C is a given constant
greater than 1.
We have, by C.5.3),
N °°
?(«) = y i-^4-5 f Mz^±i^_^-
-s
The sum
is of the form considered in the above lemma, with g(u) ~ u~a, and
t
flu) = -^, f'(u) =
Thus
Hence
ns J us^ v ;
X
JSfl-s_xl-s
Hence ?(«) = > ---—:-f O^-^-l-OI-^—-
Making N ->cc, the result follows.
| Hardy and Littlewood C).

78 APPROXIMATE FORMULAE Chap. IV
4.12. For many purposes the sum involved in Theorem 4.11 contains
too many terms (at least A\t\) to be of use. We therefore consider the
result of taking smaller values of x in the above formulae. The form
of the result is given by Theorem 4.9, with an extra factor g(n) in the
sum. If we ignore error terms for the moment, this gives
erl«* J e g(xv).
Taking .,
/ \ n ?, \ tiOSU
g(u) = u-°, f(u) = -—§_,
and replacing a, b by x, N, and i by —i, we obtain
e-2iri{(ll2ir)log(ll2irv)-(tl2n)}
2 1
—
Now the functional equation is
where x(s) = 2s-17rssec|57r/rE).
In any fixed strip a^a^jS, asf->oo
0 = (o+it—|)log(iO—i<+|log27r+o(-V D.12.1)
Hence I>+^) = r+^-ie-i^-^+il'^-i)BTr)i{l +o[i]}, D.12.2)
x ^p{()) D.12.3)
Hence the above relation is equivalent to
Z, ns AW Z/ vi-s
x<n<iV t/27rN<v^t/27rx
The formulae therefore suggest that, with some suitable error terms,
««> ~ 2 h
where 2-nxy = |<|.

4.12 APPROXIMATE FORMULAE 79
Actually the result is that
2 ~r-s+°(x-a)Jr°(Wi-aya-1) D-12-4)
for 0 < a < 1. This is known as the approximate functional equation.^
4.13. Theorem 4.13. If h is a positive constant,
0 < a < 1, 27T-a;# = ?, ?>&>(), y > h > 0,
then
n
This is an imperfect form of the approximate functional equation in
which a factor log|2| appears in one of the 0-terms; but for most
purposes it is quite sufficient. The proof depends on the same principle
as Theorem 4.9, but Theorem 4.9 would not give a sufficiently good
0-result, and we have to reconsider the integrals which occur in this
problem. Let t > 0. By Lemma 4.10
N
and the last term is O(x-alogt). If 2ttNt] > t, the first term is v = 0, i.e.
N
C du _ Ni-o—x1-*
J Us~ 1—5
Hence by D.11.2)
N
f
I
plirivu
since x1-8/{l—s) = 0(x-a) = O{x-°logt).
00
Now - du = T(l-s)l^] ,
J us \ i /
0
and by Lemma 4.3
J
= 01
\v— {tflTrJS )) \ v
N
x
[U?~~S ~\X ^dTTtV C
e2mvu\ _
1—-5 Jo 1—-SJ
0 0
J
f Hardy and Littlewood C), D), F), Siegel B).

80
Hence
APPROXIMATE FORMULAE
Chap. IV
2
N
s-1
There is still a possible term corresponding to y— rj < v
by Lemma 4.5, x
for this,
J
giving a term
Finally we can replace v ^ y—r] by v ^. y with error
Also for t > 0
0
*(«) =
Hence the result follows on taking JV large enough.
It is possible to prove the full result by a refinement of the above
methods. We shall not give the details here, since the result will be
obtained by another method, depending on contour integration.
4.14. Complex -variable methods. An extremely powerful method
of obtaining approximate formulae for ?(s) is to express ?(s) as a contour
integral, and then move the contour into a position where it can be
suitably dealt with. The following is a simple example.
Alternative proof of Theorem 4.11. We may suppose without loss of
generality that x is half an odd integer, since the last term in the sura,
which might be affected by the restriction, is O(x~a), and so is the
possible variation in x1~sj{l—s).

4.14 APPROXIMATE FORMULAE 81
Suppose first that a > 1. Then a simple application of the theorem
of residues shows that
X + ioo
= — 7T f
2l J
n<x n>x
x—ioo
x + ico
If If x1-11
= — —. (cotnz—i)z~s dz — — (cot77Z-fiJ-srf2 .
2,1 J 2,i J 1—$
x — ioo x
The final formula holds, by the theory of analytic continuation, for all
values of s, since the last two integrals are uniformly convergent in any
finite region. In the second integral we put z = x-\-ir, so that
and \z~s\ = \z\-aet&Tez < x~c
Hence the modulus of this term does not exceed
X~a I g—27rr+|?r/z fa ——
J
o
A similar result holds for the other integral, and the theorem follows.
It is possible to prove the approximate functional equation by an
extension of this argument; we may write
— COt 772 — i = 2i
Proceeding as before, this leads to an (9-term
= 0
/
x~a
2{n+lOr—\t\/x)'
and this is O(x~a) if 2(n-\-lOr—\t\/x > A, i.e. for comparatively small
values of x, if n is large. However, the rest of the argument suggested
is not particularly simple, and we prefer another proof, which will be
more useful for further developments.
4.15. Theobem 4.15. The approximate functional equation D.12.4)
holds for 0^.a^.l,x>h>0, y > h > 0.
It is possible to extend the result to any strip —k<o<kby slight
changes in the argument.
For a > 1 Us) =

82 APPROXIMATE FORMULAE Chap. IV
Transforming the integral into a loop-integral as in § 2.4, we obtain
1 e-iiraT{l—s) f ivs-1e~mw
+J
1YL
nt 1
= 2 ?
n =
f ivs-1e~m
where 0 excludes the zeros of ew— 1 other than w = 0. This holds for
all values of s except positive integers.
Let t > 0 and x ^ y, so that a: ^ J(t/27r). Let cr ^ 1,
We deform the contour C into the straight lines C1} C2, Cz, O4 joining oo,
C7]-\-irj(l-\-c), —C7]-\-ir](l—c), — cr)—Bq~\-lOri, oo, where c is an absolute
constant, 0 < c ^ ^. If y is an integer, a small indentation is made
above the pole at w = i-q. We have then
'Ci
ct
Let iv = u+iv = pe1* @ < <f> < 2n). Then
On O
4,
J
Cx
|tt, /> > ^4^, and |ew
(CO
v<j-le-lTrt f e-m
—1| > ^4. Hence
-C7]
On O
3,
"""+arctan
X 0
>
where A > 0, since
arctan 0 =
_ f d[L
J T
d[i
0
Hence
and \ew—1| > ^4. Hence
/-
On Ci, |ew—1| > ^4eM. Hence
= o\va-
J
'—1
—< arctan
Since m+1
= tj-q, and
u
l

415 APPROXIMATE FORMULAE 83
we have
<i±^ * > arctan
U 7] C '
c
= ^7T+c—arctan- = \tt-\-A,
since for 0 < 6 < 1 9
arctan 6 < —tt— = .
o
Hence
f s~i( i f /i ^w t \ / r° \
= (JIt)"-1 e-B7r+A>t du\-\-O\r]a-1 e~xu du)
d \ i I \ 4 /
Finally consider O2. Here w = iry+Aei»7r, where A is real, |A|
Hence
-1 = exp[(s—1){^'
r (
= exp (s—1I h
L I
Also
p -mw +xw /p(x-m~ l)u\ I Ax -m)u\
(f*<0),
which is bounded for u < — \n and % > \n; and
Hence the part with \u\ > ^77 is
ctjV2
-C7jV2
ICO x
, c 1
-i j
The argument also applies to the part \u\ ^ \Tr\?\ew—\\ > A on this
part. If not, suppose, for example, that the contour goes too near to
the pole at w = 2qni. Take it round an arc of the circle \w—2q-ni\ = ^n.
On this circle,
w

84 APPROXIMATE FORMULAE Chap. IV
and
iO
te
Since
this is —fat+{s—l)logBq7r) + 0A).
Hence l^3-^-™™! = 0{qa-1e~z7Tl).
The contribution of this part is therefore
Since e~insF{l— s)
we have now proved that
n=l n=l
The 0-terms are
This proves the theorem in the case considered.
To deduce the case x ^ y, change 5 into 1—5 in the result already
obtained. Then
Multiplying by x{$), and using the functional equation and
we obtain ^(.s) - x(.s)
Interchanging x and y, this gives the theorem with x ^ y.
4.16. Further approximations.! A closer examination of the
above analysis, together with a knowledge of the formulae of § 2.10,
shows that the 0-terms in the approximate functional equation can be
replaced by an asymptotic series, each term of which contains trigono-
trigonometrical functions and powers of t only.
t Siegel B).

4.16 APPROXIMATE FORMULAE
We shall consider only the simplest case in which x = y
rj = JBnt). In the neighbourhood of w = it] we have
85
irj
irj
Hence we write
e(s~l)log(.wlir])
where
#0 =
n 0
say. Now
df ls\ \ a—
Hence B+ \'t) J nan zn~x = (a— 1 +iz2) J an zn,
and the coefficients an are determined in succession by the recurrence
formula
(nJrl)\'t.an+1 = (a—n—l)an-{-ian_2 (n = 2, 3,...),
this being true for n = 0, n — 1 also if we write a_2 = a_x = 0. Thus
«o — > ai = '~3f' a*= 2t '
It follows that aH = O(r*"+[J"J) D.16.1)
(not uniformly in n); for if this is true up to n, then
Hence D.16.1) follows for all n by induction.
Now let <f>(z) = J anz-7^B).
n=0
Then ^ =
If
= —.
2ttz J
J wn{w—z)

86 APPROXIMATE FORMULAE Chap. IV
where F is a contour including the points 0 and z. Now
Hence for \w\ < §V? we have
5 6 V?
Let |z| < |V?, and let F be a circle with centre w == 0, radius /o^, where
< H-
Then ^(z) =
The function /j-^eW^has the minimum Ee/2N^N for p = BN«Jt/5)$;
pN can have this value if
Hence
27
50 '
5 /
For fz| ^ j^Jt we can also take pN = H\z\, giving
Now consider the integral along G2, and take c = 2~t. Then
J ew—\ J v "
ew—l
n
* nUVBir)
dw4-
c2
If le1*—1| > A on C2, the last integral is, as in the previous section,
T-lrt-ivl ,
0 AINJt)*
= 0
for iV < ^4L The case where the contour goes near a pole gives a similar
result, as in the previous section.

4.16 APPROXIMATE FORMULAE 87
In the first N terms we now replace G2 by the infinite straight line
of which it is a part, C'2 say. The integral multiplying an changes by
1°°
va-le-lirt f e-(A2/477)+G?A/27rV2)
J
l
-(m+lXA/v2)
W
Since ra-f-1 > tjrj — rjlBn), this is
We can write the integrand as
p-X2l8n v r-X-l8
and the second factor is steadily decreasing for A > 2*J(mr), and so
throughout the interval of integration if n < N < At with A small
enough. The whole term is then
Also an = (rn-rn+l)z-» =
Hence the total error is
l^\kn\ L
V
n = 0 \ / / I n = 0 \ / J
Now (t/n)^n increases steadily up to n = tje, and so if n < At, where
A < 1/e, it is Qf^iAiogiuy
Hence if N < ^4^, with ^4 small enough, the whole term is
We have finally the sum
Z-, inBrr)^n J
ew-l
The integral may be expressed as
(W~irj)ndw.
expj — {w-\-2m7ri—ir]J-\--^~(w-{-2nnTi—irj)—mw x
(w4-2miri—iv)n T
X v—! LL. dw,
ew—l
where L is a line in the direction arg w = \n, passing between 0 and 2tti.

88 APPROXIMATE FORMULAE Chap. IV
This is n\ times the coefficient of ?n in
— exp — (w-\-2miri — irj)
L
J
where ?(«) =
cosTra
= 9tt( 1 \m-lp-^it-(bi7Tl&)\\J'l V <>m\t\p\it
\tt
0 v /r^
Hence we obtain
^C ^ n.iv~n
Denoting the last sum by SN, we have the following result.
Theorem 4.16. // 0 < a < 1, m = [J(t/2ir)]f and N < At, where A
is a sufficiently small constant,
771 i
771 i 771 i
^ 2 ^
71 = 1 71 = 1
)™-ie-h*s-VBrrt)is-h-?i-(™lvr{l -s)lsN+oil— \
4.17. Special cases
In the approximate functional equation, let a = \ and
Then D.12.4) gives
*0 2 »~*+tf+O(ri). D.17.1)

4.17 APPROXIMATE FORMULAE 89
This can also be put into another form which is sometimes useful.
We have
so that Ix(i+*OI =
Let
so that
Let Z(t) = e*H(l+it) = {x(*+*0}-K(i+»0. D.17.2)
Since
we have also Z«) = -2^^^^. D.17.3)
The function Z(t) is thus real for real t, and
Multiplying D.17.1) by e**, we obtain
= 2 J w-icos(^—t logn) + O(t-z). D.17.4)
Again, in Theorem 4.16, let N = 3. Then
,
and the O-term gives, for ?(s), a term O(^~iff~i). In the case a = |we
obtain, on multiplying by ei9 and proceeding as before,
m
Z(t) = 2 2
n=l

90
APPROXIMATE FORMULAE
Chap. IV
4.18. A different type of approximate formula has been obtained by
Meulenbeld.f Instead of using finite partial sums of the original
Dirichlet series, we can approximate to ?(s) by sums of the form
n3
where <f>(u) decreases from 1 to 0 as u increases from 0 to 1. This reduces
considerably the order of the error terms. The simplest result of this
type is
t(s) = 2 > f
ns
nx
V<n<2y
valid for 2-nxy = \t\, \t\ >
n WxHxHy
y<n<2v v '
, — 2 < a < 2.
There is also an approximate functional equation^ for (?(<s)}2. This is
mr=y
where 0 < a ^ 1, xy = (?/2ttJ, x
this are rather elaborate.
y ?§!) + O(x*-fflogO, D.18.1)
> 0, y ^ h > 0. The proofs of
NOTES FOR CHAPTER 4
4.19. Lemmas 4.2 and 4.4 can be generalized by taking Ftobek times
differentiate, and satisfying |F(/e)(x)| ^ X > 0 throughout [a, 6]. By
using induction, in the same way that Lemma 4.4 was deduced from
Lemma 4.2, one finds that
6
The error term O(X2 ^3 i) in Lemma 4.6 may be replaced by
^x Xf), which is usually sharper in applications. To do this one
chooses 5 = ^3~? in the proof. It then suffices to show that
-5
D.19.1)
if / has a continuous first derivative and satisfies f(x) <^ x3d~3,
f Meulenbeld A).
% Hardy and Littlewood F), Titchmarsh B1).

4.19 APPROXIMATE FORMULAE
f'(x) < x25~3. Here we have written X = %F"{c) and
fix) = F(x + c) - F(c) - %x*F"{c).
If 5 ^ (X5)-1 then D.19.1) is immediate. Otherwise we have
* -(XS)~l (/l<5)-i d
91
The second integral on the right is trivially 0{(X5)~1}, while the third,
for example, is, on integrating by parts,
.a^'nx>-
2iXx
/
dx\ 2iXx
max
Xx
— 1
2iXx2
dx
2iXx2
dx
as required. Similarly the error term O {(b — a)X \X |} in Theorem 4,9 may
be replaced by 0{{b — a)X\}.
For further estimates along these lines see Vinogradov [2; pp. 86-91 ]
and Heath-Brown [11; Lemmas 6 and 10]. These papers show that the
error term O((b — a)X\x^) can be dropped entirely, under suitable
conditions.
Lemmas 4.2 and 4.8 have the following corollary, which is sometimes
useful.
Lemma 4.19. Let f(x) be a real differentiable function on the interval
[a, 6], let f'(x) be monotonic, and let 0 < X ^ | f'{x)\ ^ 3 < 1. Then
a < n < b

92 APPROXIMATE FORMULAE Chap. IV
4.20. Weighted approximate functional equations related to those
mentioned in §4.18 have been given by Lavrik [1 ] and Heath-Brown [3;
Lemma 1], [4; Lemma 1]. As a typical example one has
\y/
D.20.1)
uniformly for t ^ 1, \a\ ^ \t, xy = {t/2n)k, x, y ^> 1, for any fixed positive
integer k. Here
c- i
(c > max@, — it)).
The advantage of D.20.1) is the very small error term.
Although the weight ws(u) is a little awkward, it is easy to see, by
moving the line of integration to c = + 1, for example, that
wAu) =
uniformly for O^o-^l, < ^ 1. More accurate estimates are however
possible.
To prove D.20.1) one writes
c+i«>
v-i 7 / x ln \ 1 I (,-> v i r(A-(s + 2:)} „, \* ,*dz
YtdhijC)n-*wA--\ = — I ftO"** pn v C(^ + ^) **e* —
(c > max@, l—o-)),
and moves the line of integration to R(z) = —d,d> max@, a), giving
\k Hz
:(8 + z)\ x*e* —
The residue term is easily seen to be O{xl-a log*B + x)e~t2/4}. In the
integral we substitute z= —w, x = (t/2n)ky-1, and we apply the

4.20 APPROXIMATE FORMULAE 93
functional equation B.6.4). This yields
- d + i <*
2ni
as required.
Another result of the same general nature is
m>n = 1
D.20.2)
for < ^ 1, and any fixed positive integer &, where
^« J T r{i(i + io)r{id-io} j «•
l_joc
This type of formula has the advantage that the cross terms which
would arise on multipling D.20.1) by its complex conjugate are absent.
By moving the line of integration to RB) = +? one finds that
and Wt(u) = O(u ~k) for u ^ 1. Again better estimates are possible. The
proof of D.20.2) is similar to that of D.20.1), and starts from the formula
l_jGO
,2^
Z

94 APPROXIMATE FORMULAE Chap. IV
4.21. We may write the approximate functional equation D.18.1) in
the form
The estimate R(s, x) -^ x^~a\ogt has been shown by Jutila (see Ivic [3;
§4.2]) to be best possible for
t
x —
2n
Outside this range however, one can do better. Thus Jutila (in work to
appear) has proved that
/ x\
R(s}x)^^x-a(\ogt)\og 1+- +*-1 *!-"(/ +log*)
\ * /
for 0 ^ a ^ 1 and x > tf> 1. (The corresponding result for x -4 t may be
deduced from this, via the functional equation.) For the special case
x = y = t/2 n one may also improve on D.18.1). Motohashi [2], [3], and in
work in the course of publication, has established some very precise
results in this direction. In particular he has shown that
where A(x) is the remainder term in the Dirichlet divisor problem (see
§ 12.1). Jutila, in the work to appear, cited above, gives another proof of
this. In fact, for the special case a = i, the problem was considered 40
years earlier by Taylor A).

THE ORDER OF ?(*) IN THE CRITICAL STRIP
5.1. The main object of this chapter is to discuss the order of ?(s) as
?->oo in the 'critical strip' 0 < o < 1. We begin with a general dis-
discussion of the order problem. It is clear from the original Dirichlet
series A.1.1) that ?(<s) is bounded in any half-plane a ^ 1 + 8 > 1; and
we have proved in B.12.2) that
Us)=O(\t\) (a>i).
For o < \, corresponding results follow from the functional equation
In any fixed strip a < a < /?, as t -> oo
by D.12.3). Hence
1(8) = 0{t\-a) (a<-8<0), E.1.1)
and l(s) = 0{t%+*) (a > -8).
Thus in any half-plane a ^ cr0
l{s) = O(|*|*), k = k(a0),
i.e. ?(<s) is a function of finite order in the sense of the theory of Dirichlet
series. |
For each a we define a number fi(a) as the lower bound of numbers ?
such that yi i '*\ r\i\*\P\
It follows from the general theory of Dirichlet series % that, as a function
of a, /jl(cf) is continuous, non-increasing, and convex downwards in the
sense that no arc of the curve y = /jl(cf) has any point above its chord;
also fi(a) is never negative.
Since ?(<s) is bounded for a ^ 1 + 8 (8 > 0), it follows that
^(a) = 0 (a > 1), E.1.2)
and then from the functional equation that
M(a) = \-a (a < 0). E.1.3)
These equations also hold by continuity for a = 1 and a = 0 respec-
respectively.
t See Titchmarsh, Theory of Functions, §§ 9.4, 9.41.
% Ibid., §§ 5.65, 9.41.

96 THE ORDER OF C(s) Chap. V
The chord joining the points @, \) and A, 0) on the curve y = /n(a) is
y = \—\o. It therefore follows from the convexity property that
/*(*) < \-h @ < a < 1). E.1.4)
In particular, /n(|) < \, i.e.
?(*+»*) = O(t^) E.1.5)
for every positive e.
The exact value of fi(a) is not known for any value of a between 0
and 1. It will be shown later that fi(^) < |, and the simplest possible
hypothesis is that the graph of fi(a) consists of two straight lines
fi(a) = \-o (a < i), 0 (a > \). E.1.6)
This is known as Lindelof's hypothesis. It is equivalent to the state-
ment that
E.1.7)
for every positive e.
The approximate functional equation gives a slight refinement on the
above results. For example, taking a = \, x = y = ^{t^ir) in D.12.4),
we obtain
2
). E.1.8)
5.2. To improve upon this we have to show that a certain amount
of cancelling occurs between the terms of such a sum. We have
b b
n-s =
n~ae-inoSn
n=a+l
n=a+l
and we apply the familiar lemma of 'partial summation'. Let
b1>b2^ ... > bn > 0,
and sm = a1+a2+...+am
where the a's are any real or complex numbers. Then if
\sm\ ^M (m = 1, 2,...),
\a1b1+a2b2+...+anbn\ < Mbv E.2.1)
For
a161+. ..+anbn = b1s1+b2{s2—s1)-\-...-\-bn{sn—sn^)

5.2 IN THE CRITICAL STRIP 97
Hence
|o161+...+on6B| < JflfF1-62+... + 6B_1_6B+6B) = Mb,.
If 0 < bx < b2 < ... < bn we obtain similarly
|a161+...+aB6n| < 2Mbn.
If aB = e-«iogn5 ^ = n-a} where a > 0, it follows that
2 n-s = O(a~a max 2 e-«iogn \ E.2.2)
n = a + l ^ a<c=^6 n=a+l '
This raises the general question of the order of sums of the form
2=2 elni^n\ E.2.3)
l
2
when/(n) is a real function of n. In the above case,
The earliest method of dealing with such sums is that of Weyl,|
largely developed by Hardy and Littlewood.% This is roughly as follows.
We can reduce the problem of S to that of
S =
where gr(n) is a polynomial of sufficiently high degree, say of degree k.
I ?12 __ y y e27ri{g(?n)-g(n)} __ y y e277i{fir(n+v)-fir(n)}
v n
g2iri{ff(»+v)-B(n)}| E.2.4)
m n v n
with suitable limits for the sums; and g(n-\-v)—g(n) is of degree k— 1.
By repeating the process we ultimately obtain a sum of the form
b
?*= 2
We can now actually carry out the summation. We obtain
e2ni(b-a)X l
^ - l_e*r*A ^ |sin7rA|
If | cosec ttA| is small compared with b—a, this is a favourable result, and
can be used to give a non-trivial result for the original sum S.
An alternative method is due to van der Corput.§ In this method we
approximate to the sum S by the corresponding integral
t Weyl A), B). t Littlewood B), Landau A5).
§ van der Corput (l)-G), van der Corput and Koksma A), Titchmarsh (8)-A2).

98
THE ORDER OF C(s)
Chap. V
and then estimate the integral by the principle of stationary phase, or
some such method. Actually the original sum is usually not suitable
for this process, and intermediate steps of the form E.2.4) have to
be used.
Still another method has been introduced by Vinogradov. This is in
some ways very complicated; but it avoids the k-fold repetition used
in the Weyl-Hardy-Littlewood method, which for large k is very
'uneconomical'. An account of this method will be given in the next
chapter.
5.3. The Weyl-Hardy-Littlewood method. The relation of the
general sum to the sum involving polynomials is as follows:
Lemma 5.3. Let k be a positive integer,
b—a _ ..
and
Then
For
m=l
t > 1,
( (m lm2
exp — itl o~2
I \ a 2 a2
a
b
i
n=a + l
ka* j
<AM.
b—a).
V e-it log n __ V e-itlog(a+m)
b—a
vi =1
b — a
m=\
m=l
CO
v = 0
say,
b — a
v = 0
m=\

5.4 IN THE CRITICAL STRIP 99
5.4. The simplest case is that of ?(^+^), and we begin by working
this out. We require the case k = 2 of the above lemma, and also the
following
Lemma. Let S =
m =
Then \S\2 < At+2^2 min(/x, |cosec
A* /*
y y
Putting m' = m—r, this takes the form
e277iBamr-ar2+jSr) <;
r=—
y g477i
m
where, corresponding to each value of r, m runs over at most fi consecu
tive integers. Hence, by E.2.5),
\S\2 < 2 min(/x, |cosec27rar[)
=/x+2 2, niin(/x, [cosec 27rar[).
r=l
5.5. Theorem 5.5. ?(?+#) = O(«ilogf«).
Let 2^3 ^ a < ^4^, 6 < 2a, and let
fj. = [Jar*]. E.5.1)
Then
6 a+/i a + 2u 6
2=2 6-«1<*»= I + I" +...+ 2 = s
n = a + l n=a + l a + /i + l a + iV^+1
where N = T—1 = ol-\ = O(^).
By § 5.3, Sv = O(itf), where M is the maximum of
TO = 1
for fif < p. By § 5.4 this is
0 M+ ^ min(iu'
cosec

1Oo
Hence
THE ORDER OF C(s)
N + l a-1
Chap. V
- ^
2 2 rai
= I v= 1 r== 1
7*— 1 y —
M,
tr
Now
which, as v varies, lies between constant multiples of tr/xfa^, or, by
E.5.1), of rift2. Hence for the values of v for which \tr\{a-\-v\i.)% lies in
a certain interval {Itt, {lzt\)ir}, the least value but one of
tr
sin
is greater than Arffi2, the least but two is greater than 2Arj^2, the least
but three is greater than SArJ/x2, and so on to O(N) = O(ti) terms.
Hence these values of v contribute
The number of such intervals {In, {l±\)n} is
Hence the v-sum is
Hence
If a = 0(^2"), the second term can be omitted. Then by partial summa
tion b
-4-t = Ofilogh) F<2a).
By adding O(logt) sums of the above form, we get

5.5 IN THE CRITICAL STRIP 101
Also
y * =ol y \) = o{t\).
/-, n-2+lt \ ?- n\) '
The result therefore follows from the approximate functional equation.
5.6. We now proceed to the general case. We require the following
lemmas.
Lemma 5.6. Let f(x) = <xxk+...
be a polynomial of degree k with real coefficients. Let
where m ranges over at most y, consecutive integers. Let K = 2k~1. Then
for & > 2
\S\K < 22Kfx.K-1+2KfxK-k 2 min(/x,
where each r varies from 1 to /x — 1. For k = 1 the sum is replaced by the
single term min(/x, |cosec7ra|).
We have
m m'
I lYh Ilh ' / 1 )
m r
u —1
wnere ^ _ y e2Tri{f(m)-f(m-ri)} __
m m
and, for each rx, m ranges over at most /jl consecutive integers. Hence
by Holder's inequality
¦+ r
where the dash denotes that the term rx = 0 is omitted. Hence
If the theorem is true for k— 1, then

102
Hence
\S\K
THE ORDER OF {(«)
Chap. V
ru...,rk-i
min(/x,|cosecGraA;!r1...rfc_1)|),
and the result for k follows. Since by § 5.4 the result is true for k = 2,
it holds generally.
5.7. Lemma 5.7. For a < b ^ 2a, k ^ 2, K = 2k~1, a = O(t), t > t
0,
S=
If a <
, then
as required. Otherwise, let
and write
a + fj. a + 2u
2
Then ?„ = O(M), where Jf is the maximum, for /x' ^ fi, of
X^
™L
By Lemma 5.6
Hence
cosec
r f
/x1-*/^ V V mi
^=1 ri,...,ri_i
minj/x,
cosec
(
N+1
cosec
by Holder's inequality.

5.7 IN THE CRITICAL STRIP 103
Now as v varies,
-1)!^...^ t{h-l)\rx...rk_x
lies between constant multiples of t{k~l)\rx...rk_x^a-k-1, i.e. of
(k— l)\rv..rk_xti-k. The number of intervals of the form {Itt,A±\)it}
containing values of \t{k~\)\rx...rk_x{a-\-vfi)-k is therefore
0{(N+l)(k-l)\rv..rk_xn-k+l}.
The part of the v-sum corresponding to each of these intervals is, as in
the previous case,
\[K—i.)'.rv..rk_xj
Hence the v-sura is
\rx-"rk-il
Summing with respect to rx,..., rk_x, we obtain
0{(N+1)/**-1 log i}+O{fMk\ogkt).
Hence
2 = O{(iV+l)M1-1/K}+O{(iV+l)iu1-1/?log1/^}+O{(iV
The first term on the right can be omitted, and since
the result stated follows.
5.8. Theorem 5.8. If I is a fixed integer greater than 2, and L = 21-1,
then ?(g) =
(a = 1-1/L). E.8.1)
The second term in Lemma 5.7 can be omitted if
a
Taking k = I and applying the result O(log0, times we obtain
E.8.2)
n < N
for iV^ t2Kl + inog1~lt. Similarly, for k < I, we find
<n

104 THE ORDER OF ?(«) Chap y
for t2l<k + 2>log~kt <N^ t2l^k + inog1-kt. The error term here is at most
O{Nl-llLta\ogH) with
/I 1\ 2 1 R__fl 1\h 1
Thus ? < 1/L. When A: = / -1 we have
_/l 2\ 2 2 2
a ~ \L ~IjT+1+TL ~ l(l +
and for 2 ^ k < / — 2 we have
a ^ \4K KJk + 2+(k + l)K~ 2{k + l)(k + 2)K^0< (TfTjZ'
It therefore follows, on summing over k, that E.8.2) holds for
N ^ tHog~lt. Hence, by partial summation, we have
X n
n ^ (t/27t)i
and the theorem follows from the approximate functional equation.
5.9. van der Corput's method. In this method we approximate
to sums by integrals as in Chapter IV.
Theorem 5.9. Iff(x) is real and twice differentiable, and
0< A2 </"(*) < hX2 (or A2 < -f"{x) < hX2)
throughout the interval [a, b], and b ^ a +1, then
2 e^fM= 0{h(b-a)\j}+0(\-i).
2
^
If A2 ^ 1 the result is trivial, since the sum is 0F—a). Otherwise
Lemmas 4.7 and 4.4 give
0{(i3-a4-l)A2-i}+0{log(J8-a+2)},
where jS-a =f'(a)-f'{b) = 0{{b-a)hX2}.
Since log(jS-a+2) = O(j3-a+2) = 0{(b-a)h\.z}+0{l)
= 0{{b-a)h\\}+0{\),
the result follows.

5.10
IN THE CRITICAL STRIP
5.10. Lemma 5.10. Let f(n) be a real function, a < n
a positive integer not exceeding b—a. Then
105
6, and q
2
e27ri{/(n+r)-/(re)}
—r
For convenience in the proof, let e2nif{n) denote 0 if n ^ a or n > 6.
Then q
y e2nif(n) = ? "V "V e2rrif(m+n)
n q Zv ?-i
n m = l
the inner sum vanishing if n ^ a—g orn > 6 — 1. Hence
2
n
1
771=1
771=1
I*
- n n
Since there are at most b—a-\-q ^ 2F—a) values of n for which the
inner sum does not vanish, this does not exceed
Now
n 7n = l
e2nif(m+n)
771 = 1
2 a a
a a
— V V g27ri"{/(/n+w)-/C/Li+n)}
1 1
771 = 1 /^ =
Hence
n
<7
771=1
2F—a
m<ij.
y y y e2ni{f(m-
In the last sum, /(m+n)-/(fi+n) =f(v-\-r)—f(v), for given values
of v and r, 1 < r ^ ^ — 1, just q—r times, namely /jl = 1, m = r+1, up
to fx = q—r, m = q, with a consequent value of n in each case. Hence
the modulus of this sum is equal to
—r) y e2iri{f{-v+r)-f{-v))
r=l
e2rri{f{v+r)-f(V)) _ E.10.1)
Hence
e2irif(n)
n
-U(b-aJq+4(b-a)qq^ ^
^ ^ r = l v
and the result stated follows.

106
THE ORDER OF C(s)
Chap. V
5.11. Theorem 5.11. Let f(x) be real and have continuous derivatives
up to the third order, and let A3 < f"'(x) < hXz, or A3 < —f'"(x) < hX3,
and b—a ^ 1. Then
Let 9(x)=f(x+r)-f(x).
Then g"{x) = f"(x+r)-f"(x) = rf"'
where x < ? < x-\-r. Hence
rX3 < g"(x) < ^rA3,
or the same for ~g"(x). Hence by Theorem 5.9
a<n<6-r
Hence, by Lemma 5.10,
2
a<n<6
^
The first two terms are of the same order in A3 if q =
that A3 < 1. This gives
provided
as stated. The theorem is plainly trivial if A3 > 1. The proof also
requires that q ^ b—a. If this is not satisfied, then b—a = O(X~i),
and the result again follows.
5.12. Theorem 5.12.
Taking/(a;) = — B7r)-^loga;, we have
Hence if 6 ^ 2a the above theorem gives

5.12 IN THE CRITICAL STRIP 107
and the second term can be omitted if a < A. Then by partial summa-
summation
,
Also, by Theorem 5.9,
and hence by partial summation
Hence E.12.1) is also true if ti < a < ?. Hence, applying E.12.1)
0(log?) times, we obtain
y 4&=
n<t
and the result follows.
5.13. Theorem 5.13. Let f(x) be real and have continuous derivatives
up to the k-th order, where k ^ 4. Let Xk < /(A)(^) < hXk (or the same
for —/<*)(»)). Letb-a > 1, X = 2*.
where the constants implied are independent of k.
If Xk > 1 the theorem is trivial, as before. Otherwise, suppose the
theorem true for all integers up to k—1. Let
g(x) = f(x+ry-f(x).
Then gV<-V(x) = fk-V(x+r)-fW(x) = rf
where x < ? < x-\-r. Hence
Hence the theorem with k—1 for k gives
—r
(writing constants ^j, ^42 instead of the O's). Hence
q-i r ai-iKK-2)
since T r-1/^-2) < I r-1/^'-2) rfr =
J
o

108 THE ORDER OF {(«) Chap y
for K ^ 4. Hence, by Lemma 5.10,
To make the first two terms of the same order in Xk, let
Then
q-VBK-4)\j-UBK-4)
and we obtain
This gives the result for k; the constants are the same for k as for k—1
if A
which are satisfied if Ax and A2 are large enough.
We have assumed in the proof that q < b—a, which is true if
1) < b~a. Otherwise
and the result again holds.
5.14. Theorem 5.14. If I > 3, L = 2'-1, a = 1— Z/Bi—2),
^E) = 0(*i/<2?-2>logO. E.14.1)
We apply the above theorem with
) = J}2Si,
' 2rr
If a < n < 6 < 2a, then
and we may apply the theorem with
_ (k-l)lt h_ k

5.14 IN THE CRITICAL STRIP 109
Hence
2>). E.14.2)
The second term can be omitted if
a < AtKKkK-2K+2\ E.14.3)
Hence by partial summation
n~s = O^-o-WK-nWK-V) E.14.4)
b
subject to E.14.3). Taking a = l — ljBL—2),
? w-» == O(ai/BZ-2)-fc/BX-2)<iyBK:-2)^ E.14.5)
a<n<6
First take k = I. We obtain
J w-s = O@2L-2)) (a < AtLl(-lL-2L+*>).
a<n^b
Hence 2 71- ^ +
E.14.6)
2 ^=2 + 2 +••••
and to each term 2 corresponds a k < Z such that
1
Then V — = 0{^{//BZ'-2)-;1:/B-s:-2)}-k:/{(A:+1)-k:-2-5:+1}+1/B-k:-2)).
1
The index does not exceed that in E.14.6) if
II k \ K 1 1
\2L—2 2K—2) (k+l)K—2K+l ] 2K—2^2L-2'
which reduces to L—K > (l—k)K,
i.e. z —i jj? t—ic
which is true. Since there are again 0(logO terms,
-^w ns
The result therefore follows. Theorem 5.12 is the particular cas:
1 = 3, L = 4.

110 THE ORDER OF ?(s) Chap. V
5.15. Comparison between the Hardy-Littlewood result and
the van der Corput result. The Hardy-Littlewood method shows
that the function /u(o-) satisfies
and the van der Corput method that
For a given k, determine Z so that
Then E.15.2) and the convexity of /u(o-) give
11 l—l
1\ 2k~1 2Z—2 1 2'-1—2
n*l J
Z-lZ2/-2^ Z —1 J2'-2
J
2'-1—2~2'—2 2'-1 — 2~2/— 2
2/-A—1 1
if (A;+l)B/-1—2*-1) ^ (I—
Since 2*-1 > B/-1-2)/(Z-l), this is true if
i.e. if yb+1 < Z—1.
Now 2*-1 < 1i ^
2 < < ^ < 2
if Z ^ 8. Hence the Hardy-Littlewood result follows from the van der
Corput result if Z ^ 8.
For 4 ^ Z ^ 7 the relevant values of 1—a are
H.-L. \, \, iq.
\t A P 2 1. _3_ _1
v . u. \y. 7, Q, 31, 18.
The values of & and Z in these cases are 3, 4, 5 and 5, 6, 7 respectively.
Hence k <, I—2 in all cases.

5.16 IN THE CRITICAL STRIP 111
5.16. Theorem 5.16.
We have to apply the above results with k variable; in fact it will
be seen from the analysis of § 5.13 and § 5.14 that the constants implied
in the O's are independent of k. In particular, taking a = 1 in E.14.4),
we have ,
y /B)) (a < 6 < 2a)
uniformly with respect to k, subject to E.14.3). If
it follows that
V 1 =
6
2 sns= 2 + 2
a<n<6
and applying the above result with k = 2, 3,..., or r, we obtain, since
there are O(logt) terms,
L E.16.1)
+
Let r = [loglogZ]. Then
2R <
and
Hence the above sum is bounded. Also
2
\ r j \loglog^
This proves the theorem.
The same result can also be deduced from the Weyl-Hardy-Little-
wood analysis.

112 THE ORDER OF {(*) Chap. V
5.17. Theorem 5.17. For t > A
= O(log*), CT>1_<^«, E.17.1)
* E.17.2)
M some At), and ^L^ = ojJSSi.), E.17.3)
«y = %$#)¦ <5-17-4>
We observe that E.14.1) holds with a constant independent of I, and
also, by the Phragmen-Lindelof theorem, uniformly for
a ^ 1— 1/{2L—2).
Let t be given (sufficiently large), and let
Then L < 2A/log2)log(log'/loglog')-1 = &
"^ 2 loglog ^'
and similarly L ^ -:;—^— .
4 loglog t
Hence
Z Z ^ loglog^—logloglog^—log 2 loglog* ^ (loglogtf
log 2 log* ^ log*
for * > A (large enough). Hence if
>1 .
then a ^ 1 —
Hence E.14.1) is applicable, and gives
log*
= O(log5*).
This proves E.17.1). The remaining results then follow from Theorems
3.10 and 3.11, taking (for * > A)
«

5.18 IN THE CRITICAL STRIP 113
5.18. In this section we reconsider the problem of the order of
?(i-H0- Small improvements on Theorem 5.12 have been obtained by
various different methods. Results of the form
.,, 163 27 229 19 15
with a = , , , , —
988' 164 1392' 116' 92
were proved by Walfisz A), Titchmarsh (9), Phillips A), Titchmarsh
B4), and Min A) respectively.! We shall give here the argument which
leads to the index ^. The main idea of the proof is that we combine
Theorem 5.13 with Theorem 4.9, which enables us to transform a given
exponential sum into another, which may be easier to deal with.
Theorem 5.18.
Consider the sum
i ~~
where a < b ^ 2a, a < A*lt. By § 5.10
SM(f
where q ^ b—a, and
2 = y e-fl{log(n+r)-lo8rn}_
—r
We now apply Theorem 4.9. to S2. We have
J K ' r
We can therefore apply Theorem 4.9 with A2 = tra~3, A3 = tra,-*. Thus
E.18.2)
where <f>(v) ~ f(xv)—vxv, a = f'(b—r), j3 = /'(a). Actually the log-term
can be omitted, since it is 0(?r!a-5).
Consider next the sum
^ (a < y < j8).
I Note that the proof of the lemma in Titchmarsh B4) is incorrect. The lemma
should be replaced by the corresponding theorem in Titchmarsh A6).

114 THE ORDER OF {(*)
The numbers xv are given by
tr . 1/
= v, i.e. xv = -lr*
Hence
Chap. V
7TV
7
(I 2\
77T/
since
It follows that
rf'(a) =
2tt
where Kv K2,..., and
Theorem 5.13, with h
Hence
l< ((>y,
^fc depend on A; only. We may therefore apply
= 0A), and
ltr\i-m
Also l/"^)!"* is monotonic and of the form 0(t~w~2a%). Hence by
partial summation
) 2
Hence
l fl^
- > |
* r=l
-2^^ 0{(tq)~M
Inserting this in E.18.1), and using the inequality
we obtain
The first two terms on the right are of the same order if
g _ raCK-2k-
and they are then of the form

5.18 IN THE CRITICAL STRIP 115
For k = 2, 3, 4, 5, 6,..., the index has the values
13 5 17 73
2' 7' 12' IP 176""
and of these \l is the smallest. We therefore take k = 5,
q = [at^~H] (a > iH),
and obtain
This also holds if q > 6—a, since then
Sx = 0F-a) = 0(q) =
which is of smaller order than the third term in the above right-hand
side.
It is easily seen that the last two terms are negligible compared with
the first if a — 0{*Jt). Hence by partial summation
{a>tU).
Applying this with a = N, b = 2N—1; a = 2N, b = 4iY—1,... until
b = [A<Jt], we obtain
V -J-r. = 0{tTh
<n^A-Jt
= O(tN?) {N > tb).
We require a subsidiary argument for n < trr, and in fact E.14.2) with
k = 4 gives
2 ^A (a
and by adding terms of this type as before
The result therefore follows from the approximate functional equation.
NOTES FOR CHAPTER 5
5.19. Two more completely different arguments have been given,
leading to the estimate ... . ._ ir. 1X
H(b) ^ i E.19.1)

116
THE ORDER OF
Chap. V
Firstly Bombieri, in unpublished work, has used a method related to
that of §6.12, together with the bound
o o
to prove E.19.1). Secondly, E.19.1) follows from the mean-value bound
G.24.4) of Iwaniec [1]. (This deep result is described in §7.24.)
Heath-Brown [9] has shown that the weaker estimate /zQ) ^ ^
follows from an argument analogous to Burgess's [1] treatment of
character sums. Moreover the bound /iQ) ^ -fa, which is weaker still,
but none the less non-trivial, follows from Heath-Brown's [4] fourth-
power moment G.21.1), based on Weil's estimate for the Kloosterman
sum. Thus there are some extremely diverse arguments leading to non-
trivial bounds for
5.20. The argument given in §5.18 is generalized by the 'method of
exponent pairs' of van der Corput A), B) and Phillips A). Let s, c be
positive constants, and let fF{s, c) be the set of quadruples (N, I, f, y) as
follows:
(i) N and y are positive and satisfy yN~s ^ 1,
(ii) J is a subinterval of (N, 2N],
(iii) / is a real valued function on /, with derivatives of all orders,
satisfying
dxn
(yx~s)
E.20.1)
for n $s 0.
We then say that (p, q) is an 'exponent pair' ifO^p^^^g^l and if
for each s > 0 there exists a sufficiently small c = c(p, q, s) > 0 such that
exp{27ti/(n)}
nel
E.20.2)
uniformly for (N, I, f, y)e&{s, c).
We may observe that yN~s is the order of magnitude of f'{x). It is
immediate that @, 1) is an exponent pair. Moreover Theorems 5.9, 5.11,
and 5.13 correspond to the exponent pairs (^, ^), (^, f), and

5.20 IN THE CRITICAL STRIP 117
By using Lemma 5.10 one may prove that
x / P P + q + l\
A(p,q) = [ ,
\2p + 2 2p + 2 )
is an exponent pair whenever (p, q) is. Similarly from Theorem 4.9, as
sharpened in §4.19, one may show that
is an exponent pair whenever (p, q) is, providing that p + 2q ^ |. Thus
one may build up a range of pairs by repeated applications of these A
and B processes. In doing this one should note that B2(p, q) = (p, q).
Examples of exponent pairs are:
B@,1) = (J, J), AB@,1) = (?, |), A*B@,1) = to, ft),
^), ABA2.6A, 0) = (^., if),
= (if, if), ABA3?@,1) = (ft, jj),
To estimate the sum Z: of §5.18 we may take
AW-^ log., j- = ij, .-1,
so that E.20.1) holds for any c ^ 0. The exponent pair (ft, ||) then yields
whence
a <n ^ 6
for a <$ $. We therefore recover Theorem 5.18.
The estimate /i(i) ^ j^%- of Phillips A) comes from a better choice of
exponent pair. In general we will have
providing that q ^p + \. Rankin [1] has shown that the infimum of
J(P + Q — b), over all pairs generated from @,1) by the A and B processes,
is 0-16451067.... (Graham, in work in the course of publication, gives
further details.) Note however that there are exponent pairs better for

118 THE ORDER OF ?<*) Chap. V
certain problems than any which can be got in this way, as we shall see
in §§6.17-18. These unfortunately do not seem to help in the estimation
of
5.21. The list of bounds for //Q) may be extended as follows.
= 0-164979... Walfisz A),
= 0164634... Titchmarsh (9),
= 0164511... Phillips A)
0164510... Rankin [1]
= 0-163793... Titchmarsh B4)
= 0-163043... Min A)
= 0-162162 ... Haneke [1 ]
= 0162136... Kolesnik [2 ]
= 0-162037... Kolesnik [4]
= 0-162004... Kolesnik [5 ].
The value ^- was obtained by Chen [1 ], independently of Haneke, but a
little later.
The estimates from Titchmarsh B4) onwards depend on bounds for
multiple sums. In proving Lemma 5.10 the sum over r on the left of
E.10.1) is estimated trivially. However, there is scope for further savings
by considering the sum over r and v as a two-dimensional sum, and using
two dimensional analogues of the A and B processes given by Lemma
5.10 and Theorem 4.9. Indeed since further variables are introduced
each time an A process is used, higher-dimensional sums will occur.
Srinivasan [1] has given a treatment of double sums, but it is not clear
whether it is sufficiently flexible to give, for example, new exponent
pairs for one-dimensional sums.

VI
VINOGRADOV'S METHOD
6.1. Still another method of dealing with exponential sums is due to
Vinogradov.f This has passed through a number of different forms of
which the one given here is the most successful. In the theory of the zeta-
function, the method gives new results in the neighbourhood of the line
a = 1.
Let f(n) = otknk + ... + oc1n-t-oc0
be a polynomial of degree k ^ 2 with real coefficients, and let a and q
be integers,
,l) = \ ...\\S{q)\»da1..d«k.
0 0
The question of the order of J(q, I) as a function of q is important in
the method.
Since S(q) = O(q) we have trivially J(q, I) = Oiq21). Less trivially,
we could argue as follows. We have
{S(q)}k= 2 «
ni,...,nk
\S{q)\2k=
mi,...,n*
On integrating over the k-dimensional unit cube, we obtain a zero factor
if any of the numbers
mj-h...+m?—n\ —...—n\ (h = 1,..., k)
is different from zero. Hence J(q, k) is equal to the number of solutions
of the system of equations
J (h = 1,..., k),
where a < mv < a-\-q, a < nv <
But it follows from these equations that the numbers nv are equal (in
some order) to the numbers mv. Hence only the mv can be chosen
f Vinogradov UM4), Tchudakoff (l)-E), Titchmarsh B0), Hua A).

120 VINOGRADOV'S METHOD Chap. VI
arbitrarily, and so the total number of solutions is 0{qk). Hence
and J(q,l) = 0{q*~2kJ{q,k)} = 0{q*-k).
This, however, is not sufficient for the application (see Lemma 6.8).
For any integer l,.J(q,l) is equal to the number of solutions of the
equations
ro*+...+wJ = »$+•••+»* (h = 1, 2,..., k),
where a < mv ^ a-\-q, a < nv < a+g. Actually J"(g,Z) is independent
of a; for putting Mv = mv—a, Nv = nv—a, we obtain
v=l v=l
which is equivalent to
and 0 < Mv < q, 0 < Nv < g.
Clearly J(g, Z) is a non-decreasing function of q.
6.2. Lemma 6.2. Let mx,..., mk, nx,..., nk be two sets of integers, let
and let ah, a'h be the h-th elementary symmetric functions of the mv and nv
respectively. If \mv\ < q, \nv\ < q, and
K-s'h\ < q*-1 (h=l,...,k), F.2.1)
then Wh-°H\ < iPhq)*-1 (h = 2,..., k). F.2.2)
Clearly \sh\ < kq\ \s'h\ ^ kq\
and |ail <g)g*<*V.
Now a2 = i(«i—«2)«
Hence k2—^il = ilEi—S2)~ (s?—*i)l
the result stated for h = 2.
Now suppose that F.2.2) holds with h = 2,..., j— 1, where 3 < j < k,
so that j^^j ^ ^^ {h = i,..., j_i).
By a well-known theorem on symmetric functions

6.2 VINOGRADOV'S METHOD 121
Hence
,,1,
?+?«*
a = i
jfe,
since 2k/Bk— 1) < 2 and J > 3. This proves the lemma.
6.3. Lemma 6.3. Let 1 < G < q, and let gv..., gk be integers satisfying
K ft < 9. < - < gk < G, SV-SV-i > 1. F.3.1)
i^or eacA vaZwe o/ v A ^v ^. k) let mv be an integer lying in the interval
-a+(gv-l)q/G < mv < -a+gvq/G,
where 0 ^. a ^. q. Then the number of sets of such integers mv..., mkfor
which the values of sh (h = I,..., k) lie in given intervals of lengths not
exceeding qh~\ is ^ D,kG)^k{k-x).
If x is any number such that \x\ ^ q, the above lemma gives
k
\(x—m1)...{x—mk) — (z—n1)...{x—nk)\ < ? \ah—ah\\x\k-h
k h=1
since fc > 2. If nv..., nk satisfy the same conditions as mv..., mk, then
\mk—nv\ > q/G for v = 1, 2,..., fc—1. Hence, putting a: = nfe,
i.e. Nfc-nfc| < BkG)k~1.
Thus the number of numbers m^ satisfying the requirements of the
theorem does not exceed
< {4kO)k-1.

122 VINOGRADOV'S METHOD Chap. VI
Next, for a given value of mk, the numbers mv..., mk_1 satisfy similar
conditions with k — 1 instead of k, and hence the number of values of
mk_x is at most {4(fc— l)G}k~2 < {4kG)k-2. Proceeding in this way, we
find that the total number of sets does not exceed
6.4. Lemma 6.4. Under the same conditions as in Lemma 6.3, the
number of sets of integers m1(..., mkfor which the numbers sh (h = 1,..., k)
lie in given intervals of lengths not exceeding cqh{1~^k), where c > 1, does
not exceed Bc)
We divide the hth interval into
tcaha-W]
parts, and apply Lemma 6.3. Since
JJ Bcq1~hlk) = Bc)k
h
we have at most Bc)kq^k~1) sets of sub-intervals, each satisfying the
conditions of Lemma 6.3. For each set there are at most
solutions, so that the result follows.
6.5. Lemma 6.5. Let k < I, let f(n) be as in §6.1, and let
I = f ... J \Zmtgx ... ZmJ*\S(qi-Vk)\*«-» doc, ... dack,
6
6 o
where Zmgv= J e2nim
and the gv satisfy F.3.1) with 1 < G = 2m < q. Then
I < 23k-«m+Vl~Kk-V-mk{l—k)khl'*k-Uqlk-±J(q1-llk,l—k).
We have x x
7=2 W(Nv,..,Nk) f ... f e2ni(NkOCk+-+N^\S(q1-1lk)\^-k)d<x1... dock
1
W(Nv...,Nk) f ... f | S(gi-i/*)!*-»(&*! ...dock,
where ^(iV^,..., Nk) is the number of solutions of the equations
m\+...+m\—n\—...—n\ = Nh (h — 1,..., k)
for mv and nv in the interval (^v -1) 2~mq < x ^ gv2~mq. Moreover Nh
runs over those integers for which one can solve

6.5 VINOGRADOV'S METHOD 123
where m'v and n'v lie in an interval (a,a + ql-llk]. As in §6.1 we
can shift each range through -a, i.e. replace a by 0. Then Nh ranges
over at most 2(Z-&)gAlA~1/fe) values. Hence by Lemma 6.4, for given
values of rip..., nk, the number of sets of (m,,..., mk) does not exceed
Also (rip..., nk) takes not more than (l + 2-mq)k ^ Bl-mq)k values.
Hence
Nu...,Nk
and the result follows.
6.6. Lemma 6.6. The result of Lemma 6.5 holds whether the g's satisfy,
F.3.1) or not, if m has the value
\^ql F.6.1)
|Mog2j
it is sufficient to prove that
or that gifc+i < ^
or that (P-f-^)logg < \k{k+l)Mlog 2+p(&—l)log4&,
or that log g < kM log 2 + ~^- log 4Jfc.
Since JPM
this is true if k log 2 ^ -A- log 4k,
k-\-l
k—l
or log 2 <——-Iog4^,
k-f-l
which is true for k ^ 2.
6.7. Lemma 6.7. TAe set of integers (g1}..., g{), where k < I, and each gv
ranges over A, 0], is said to be well-spaced if there are at least k of them,
say gh,..., gh, satisfying
9t-9j?-x > l (v = 2— *)•
The number of sets which are not well-spaced is at most l\ 3lGk~1.
Let g[,..., g[ denote gv..., gl arranged in increasing order, and let
fv = g'v — g'vl. If the set is not well-spaced, there are at most k — 2 of
the numbers /„ for which /„ > 1.

124 VINOGRADOV'S METHOD Chap. VI
Consider those sets in which exactly h @ < h < k—2) of the numbers
/„ are greater than 1. The number of ways in which these hfv's can be
chosen from the total ^ — 1 is / 1. Also each of the hfv's can take at
\ fi )
most G values, and each of the rest at most 2 values. Since g[ takes
at most G values, the total number of sets of g'v arising in this way is
at most ii |\
The total number of not well-spaced sets g'v is therefore
k-2 ij ,x fc-2 ,j ,x
Since the number of sets gv corresponding to each set g'v is at most II, the
result follows.
6.8. Lemma 6.8. If I ^ lk*+\k and M is defined by F.6.1), then
J(q,l) < max(l,ML8^(l\)Hkk^k-1^l-^k+^-^J(q^-^,l-k).
Suppose first that M is not less than 2, i.e. that q ^ 22fc. Let /x be
a positive integer not greater than M—l. Then
- 1, i.e. 2m+i < gi/*.
2
Let ^(g)=2 I ^ = 2Z ,
say. Then ^(g)? - J, Z^ ... Z/ttW,
where each gv runs from 1 to 2m, and the sum contains 2^ terms.
We denote those products Z/Xfffl... ZM{7! with well-spaced g's by Z^.
The number of these, M^ say, does not exceed 2^. In the remaining
terms we divide each factor into two parts, so that we obtain products
of the type ?M+i>ffl... ^+i>ff;> each g lying in A,2^+!). The number of
such terms, J^+1 say, does not exceed ll&W-W = 11&2W-U, by
Lemma 6.7. The terms of this type with well-spaced g's we denote by
Z'^+1, and the rest we subdivide again. We proceed in this way until
finally Z'M denotes all the products of order M, whether containing
well-spaced ^'s or not. We then have
{8(q)Y =22 %n>
Vl=fJ.
m ^r
^M 2 I2^»I2<^ IM,nI\Z'm\2, F.8.1)

6.8 VINOGRADOV'S METHOD 125
where Mm is the number of terms in the sum 2 %'m- By Lemma 6.7,
Mm < l\ 3Wm-M-ti2l = l\ 6'2(m-iX*-i) (m > /x).
Consider, for example, 2 I^JJ2- The general Z^ can be written
7 7 7 7
where g^,..., gk satisfy F.3.1) with G = 2V-. Now, since the geometric
mean does not exceed the arithmetic mean,
1 z
v=k+l
We divide these Z^ gv into parts of length q1-1!*— l (or less). The number
of such parts does not exceed
since q1-1^ > g1/* ^ 2M > 4. Each part is of the form Siq1-^), or with
qi-ilk replaced by a smaller number. Hence by Holder's inequalityf
Hence
— Ml ^ 7 ;,
L — K
Hence by Lemma 6.5, and the non-decreasing property of J(q, I) as a
function of q,
1 i
^\Z'\2 docl... dcxk
o 6
X
A similar argument applies to Z'm, with [i replaced by m. Hence
f 2^
/1 ^ Lt
Also
^ 2(Z!J62/,
t Here S(ql~llk) denotes any sum of the form S(p) with p < g1'*.

126 VINOGRADOV'S METHOD Chap. VI
since we can start with an integer n such that 2 < /!. (Indeed we may
take n = 1.) Hence
and since 22l + k2 + k + 1621 ^ 26/62/ = 482/
the result follows.
If M < 2, i.e. q < 22Jc, divide S(q) into four parts, each of the form
S{q'), where q' < \q < q1-1^. By Holder's inequality
\S(q)\0J < 4P-1 J |%')|a <
Integrating over the unit hypercube,
and the result again follows.
6.9. Lemma 6.9. 7/r is anj non-negative integer, and I > \k2 + ±k + kr,
then
J(q, 1) < K
where 8r = P(A+l)/l -ijr, K = 482'(Z!JZ*Jti*<*-«.
This is obvious if r = 0, since then 80 = ^k(k-{-l) and J(q,l) ^ #2/.
Assuming then that it is true up to r— 1, Lemma 6.8 (in which M < log q)
gives
and the index of q reduces to 21—
6.10. Lemma 6.10. If I = [Jfc2lOg(Jfc2+Jfc) + JJfc2+«Jfc] + l, Jfc > 7,
We have 8r < Hf ^+l)[l -^j < 1,
i.e. if Iog{^+1)} < J
— 1
This is true if log{&(?+1)} < r/k,
or if r = [Jclog{k2+k)]-{-l.
Since r < Hog3&+1 < 4ifclogi*, / < F,
and
< 5ZlogZ+ZlogA: < 16ZlogA:,
the result follows.

6.11 VINOGRADOV'S METHOD 127
6.11. Lemma 6.11. Let M and N be integers, N > 1, and let <f>(n) be
a real function ofn, defined for M ^.n ^. M+N—l, such that
8 < <f>(n+l)-<f>(n) < cS {M < n < M+N-2),
where 8>0, c^jI, c8^|. Let W > 0. Let x denote the difference
between x and the nearest integer. Then the number of values of n for
which <f>(n) < WS is less than
(ATc8+l)BW+l).
Let a be a given real number, and let G be the number of values of n
for which , . . ,M ^ , 7 , s
for some integer h. To each h corresponds at most one n, so that
0 ^ h2—&J+1, where hx and h2 are the least and greatest values of h.
But clearly
4>(M) < cx+^+8, oc+h2 < 4>(M+N-1),
whence h2-hx-8 < <j>(M+N—\)—<l>(M) < {N-l)cS,
and (? < (^—1H8+8 + 1 < iVc8+l.
The result of the lemma now follows from the fact that an interval of
length 2^8 may be divided into [2W+1] intervals of length less than
8 «
6.12. Lemma 6.12. Let k and Q be integers, k > 7, Q ^ 2, and let f(x)
be real and have continuous derivatives up to the (k+l)th order in
[P+l, P + Q]; letO < X <1 and
<2A (P+l < x < P+Q) F.12.1)
or the same for —Pk+1)(x), and let
A-l < Q <A-X. F.12.2)
P + Q
Then j eimf(n) <- Aemlog3*:Q1-')logQ, F.12.3)
where p —
Let <7 =
so that 2 < q ^
and write o = 2,
T(n) = 2 e27r^"l+n)-/(n» (P+l < n < P+Q—q)-

128
Then
VINOGRADOV'S METHOD
q P + Q
m = ™?i -5+
q P+Q-q+m
y ^ e27Tif(n)
m=l n=P+l-\-m
q P+Q-q
m=ln=P+l
P+Q-q q
Chap. VI
P+Q-q
<»-!n|T(n)l+3a
F.12.4)
by Holder's inequality, where I is any positive integer.
We now write Ar — Ar(n) = /(r)(n)/r! for 1 < r < k, and define the
^-dimensional region Qn by the inequalities
Iar-Ar| ^ ^^-- (r-1,...,*). F.12.5)
If we set
5(m) = f(m + n)-f(n) — (ixkmk + ... +a1m),
then, by partial summation, we will have
However, by Taylor's theorem together with the bound F.12.1) we obtain
where 0 <&,&'< 1. If F.12.5) holds it follows that
\8'(p)\ < Xri9~r9r-1 + 3feA9* <
i
by our choice of q. We therefore have
\T(n)\
- |S(p)|rfp) =
o

6.12 VINOGRADOVS METHOD 129
say. Integrating over the region Qn, and dividing by its volume, we
obtain
L. \\S0(q)\*lda1...dak. F.12.6)
The integral of | So (q) \2l over Qn is equal to its integral taken over the
region obtained by subtracting any integer from each coordinate. We
say that such a region is congruent (mod 1) to Cln. Now let n, n' be two
integers in the interval [P + 1, P+Q—q], and let Qn, Q'n be the corre-
corresponding regions defined by F.12.5). A necessary condition that Cln
should intersect with any region congruent (mod 1) to Q!n is that
Ak(n)-Ak(n') < <r* < A?. F.12.7)
Let <f>(n) = Ak{n)-Ak{n'). Then
cf>(n+l)-cf>(n) = ^{/W(n+l)-
where n < $ < n-\-l. The conditions of Lemma 6.11 are therefore
satisfied, with c = 2 and 8 = A(&+1). Taking W = q/(k-\-l), we see
that the number of numbers n in [P + 1, P + Q — q] for which F.12.7) is
possible, does not exceed
(J^ ) ^l) < Zkq.
Since this is independent of n', it follows that
n = P+1
^3kq2*lJ(q,l), F.12.8)
since
S0(9Ji<2«-if|S(9)|« + i \\S(p)\2ldp\
o
Denning I as in Lemma 6.10, we obtain from F.12.4), F.12.6), F.12.8) and
Lemma 10
\S\ ^ 2k + 5knQ1 -hq-1 {^I«A + »SkqJ(q,
Now 9<2A-1/^ + 1)<2Q4/(ft + 1). Hence

130 VINOGRADOV'S METHOD Chap. VI
and the result follows, since ^-- 3/{(k + 1I} ^-fa- and / < Sk2 logk.
6.13. Lemma 6.13. Iff{x) satisfies the conditions of Lemma 6.12 in an
interval [P+l, P + N], where N < Q, and
X-i < Q <A-1, F.13.1)
then
e2irif(n)
F.13.2)
If A~i ^ N, the conditions of the previous theorem are satisfied
when Q is replaced by N, and F.13.2) follows at once from F.12.3).
On the other hand, if A~* > N, then
P + N
e2nif(n)
L
and F.13.1) again follows.
6.14. Theorem 6.14.
JV < A-i < Ql
Let /«-)—'-
Let a < x < b < 2a. Since ( — l)fc+1/(A:+1)(a;) is steadily decreasing, we
can divide the interval [a, b] into not more than k + 1 intervals, in each
of which inequalities of the form F.12.1) hold, where A depends on the
particular interval, and satisfies
1 %=—. F.14.1)
Let Q = a < t, log a > 2 logH and
Then Q < a*^ < Q2.
Clearly A < Q-1, while A > Q~z if Q > 2fc+27r(A:+l), or if
and this is true if t is large enough. It follows from Lemma 6.13 that
where p is defined as in § 6.12. Hence
-J— = O(ke™kl°z2ka-ologa)
n1+xi
= OJlogfexp

6.15 VINOGRADOV'S METHOD 131
Suppose that klogk < Alogm,
with a sufficiently small A, or
log a > A(\ogt\og\ogt)*
with a sufficiently large A. Then
= O[lo
and the sum of O(\ogt) such terms is bounded.
Since k ^ 7, we also require that a ^ fi. Using E.16.1) with r = 8,
and writing j8 = ^28/Gx128+1^ we obtain
The last sum is bounded if
log a = u4(log<loglogZL
with a suitable A, and the theorem follows.
6.15. IfO < cr< 1, we obtain similarly
2 ^
and this is bounded if
^(loglogQl_
1 a< logfc ~ 0>
with a sufficiently small A. Hence in this region
f— +0A)
1/
We can now apply Theorem 3.10, with
6A) = AaOff]i, W) = A logi«(loglog t)i.
Hence there is a region
F.15.1)

132 VINOGRADOV'S METHOD Chap. VI
which is free from zeros of ?(s); and by Theorem 3.11 we have also
^ = 0{log«(loglogO*}.
F.15.2), F.15.3)
NOTES FOR CHAPTER 6
6.16. Further improvements have been made in the estimation of
J(q, I). The most important of these is due to Karatsuba [2] who used a
p-adic analogue of the argument given here, thereby producing a
considerable simplication of the proof. Moreover, as was shown by
Steckin [1], one is then able to sharpen Lemma 6.9 to yield the bound
Tin 1\ ^- f^k3logk-,21 — kk(k + 1) +<5
for / ^ kr, where k ^ 2, r is a positive integer, C is an absolute constant,
and Sr = \k2 A — l/k)r. Here one has a smaller value for dr than formerly,
but more significantly, the condition / ^ \k2 + \k + kr has been relaxed.
6.17. One can use Lemma 6.13 to obtain exponent pairs. To avoid
confusion of notation, we take /to be defined on (a, b], with a < b < 2a
and X~^ a < A. Then
e2n if(n)
a <n 4 b
Now suppose that (N, I, /, y) is in the set #"(s, ^) of §5.20, whence
-s-k
with
We may therefore break up / into O(s + k) subintervals (a, b] with
b < (?I/(s + ft)a, on each of which one has
with A = J<xfta-S-*. We now choose & so that X~$ <^ JV < 2iV< A-1 for all
a in the range iV < a < 2iV. To do this we take k ^ 7 such that
JV
< fiV1-8 < —. F.17.1)
a

6.17 VINOGRADOV'S METHOD 133
Note that Nk/<xk tends to infinity with k, if N ^ 2, so this is always
possible, providing that
JV6
— <|jVi-s. F.17.2)
The estimate F.17.1) ensures that 2JV< A-1, and hence, incidentally,
that X < 1. Moreover we also have
-s
if N ^ 2s + k + 2, and so A * < iV. It follows that
F.17.3)
ne/
for N ^ 2s + * + 2, subject to F.17.2).
We shall now show that
(P, 9) =
srTi'1 o^1 F'17-4>
\(m — 2)m2 log m 25m2log my
is an exponent pair whenever m ^ 3. If yN2-s-m ^ 1 then (yN-s]PN<*
^ iV, and the required bound E.20.2) is trivial. If F.17.2) fails, then
yN-s <SN5 and, using the exponent pair (y^-, f|§-) = A5S@, 1) (in the
notation of §5.20) we have
nel
as required. We may therefore assume that yN2-s-m < 1, and that
F.17.2) holds. Let us suppose that N ^ maxBs + ™ + 2, 2(?s + l)m). Then
F.17.1) yields
l 5ss + ls + 2 s + ^-
i
4 2 3 4
((l
whence
/ 1\I \k-m
Since N ^ 2(^s + l)m we deduce that k ^ m. Moreover we then have
+ 2, so that F.17.3) applies. Since k is bounded in

134 VINOGRADOV'S METHOD Chap. VI
terms of p, q and s, it follows that
pqpq
nel
if N > p,q,s 1, and the required estimate E.20.2) follows.
6.18. We now show that the exponent pair F.17.4) is better than any
pair (a, f5) obtainable by the A and B processes from @, 1), if m ^ 106. By
this we mean that there is no pair (a, ft) with both p ^ a and q ^ j5. To do
this we shall show that
^l. F.18.1)
Then, since 5425m2 log m <{m — 2K for m ^ 106, we have q + 5p% < 1,
and the result will follow. Certainly F.18.1) holds for @,1). Thus it
suffices to prove F.18.1) by induction on the number of A and B
processes needed to obtain (a, fi). Since B2(<x, fS) — (a, /?) and A@,1)
= @, 1), we may suppose that either (a, ft) = A(y, S) or (a, ft) = BA (y, 3),
where (y, 8) satisfies F.18.1). In the former case we have
for 0 < y < \, and in the latter case
2y
for 0 < y < \. This completes the proof of our assertion.
The exponent pairs F.17.4) are not likely to be useful in practice. The
purpose of the above analysis is to show that Lemma 6.13 is sufficiently
general to apply to any function for which the exponent pairs method
can be used, and that there do exist exponent pairs not obtainable by the
A and B processes.
6.19. Different ways of using J(q, I) to estimate exponential sums
have been given by Korobov [1] and Vinogradov [1] (see Walfisz [1;
Chapter 2] for an alternative exposition). These methods require more
information about/than a bound F.12.1) for a single derivative, and so
we shall give the result for partial sums of the zeta-function only. The
two methods give qualitatively similar estimates, but Vinogradov's is
slightly simpler, and is quantitatively better. Vinogradov's result, as
given by Walfisz [1], is
? n-it<a1-p F.19.1)
a <n ^ b

619 VINOGRADOV'S METHOD 135
for a < b ^ 2a, t ^ 1, where
tllk ^a
k ^ 19, and
1
P =
60000&2'
The implied constant is absolute. Richert [3] has used this to show that
C((j + it) < A + timi~a)S (log 0* F.19.2)
uniformly for 0 < a < 2, t ^ 2. The choices
in Theorems 3.10 and 3.11 therefore give a region
a ^ 1 - A (log 0 "*(loglog t) "*
free of zeros, and in which
(log 0*(loglog 0*.
The superiority of F.19.1) over Lemma 6.13 lies mainly in the elimi-
elimination of the term expC3&2 log k), rather than in the improvement in
the exponent p.
Various authors have reduced the constant 100 in F.19.2), and the best
result to date appears to be one in which 100 is replaced by 18.8 (Heath-
Brown, unpublished).
6.20. We shall sketch the proof of Vinogradov's bound. The starting
point is an estimate of the form F.12.4), but with
Q " -¦"'-> + n)-/(n)} F.20.1)
U' V = 1
in place of T(n). One replaces f(uv + n) —f(n) by a polynomial
F(uv) = A1uv+ ... + Akukvk

136 VINOGRADOV'S METHOD
as in §6.12, and then uses Holder's inequality to obtain
i
Chap. VI
,2niF(uv)
I
, 2niF(uv)
where \t](v)\ = 1, n(o1,...,oft) denotes the number of solutions of
=a
and
...,<7fc;i;) = A1a1v+ ... +Akokvk.
Now, by Holder's inequality again, one has
,2niF(uv)
2/2
, 21-2
X
Here
and
Moreover
72niH(ov...,ok;zv..
av...,ah
where
r, ...,(Tfc: T,, ...,Tb) = A,a,X, + ...
1' J/Z*l7 7/?' Ill
and n*(ilf..., zk) is the sum of ^i^)... rf(v2l) subject to

6.20
VINOGRADOV'S METHOD
137
Since \n*(x1,...,Tk)\ ^ J(q, I), it follows that
2/2 k /
D2 n s
,27riF(uu)
At this point one estimates the sum over xh, getting a non-trivial bound
whenever q~2h <€ \Ah\ <? 1. This leads to an appropriate result for the
original sum F.20.1), on taking / = [ck2 ] with a suitable constant c. If we
use Lemma 6.9, for example, to estimate J(q, I), then
One therefore sees that the implied constant in F.19.1) is indeed
independent of k.

VII
MEAN-VALUE THEOREMS
7.1. The problem of the order of ?(s) in the critical strip is, as we have
seen, unsolved. The problem of the average order, or mean-value, is
much easier, and, in its simplest form, has been solved completely. The
form which it takes is that of determining the behaviour of
T
as T -> oo, for any given value of a. We also consider mean values of
other powers of ?(s).
Results of this kind have applications in the problem of the zeros, and
also in problems in the theory of numbers. They could also be used to
prove O-results if we could push them far enough; and they are closely
connected with the Q-results which are the subject of the next chapter.
We begin by recalling a general mean-value theorem for Dirichlet
series.
Theorem 7.1. Let
CO
ns
J V / s f*^ %J \ /
Z—• 71°
6e absolutely convergent for a > ax, a > a2 respectively. Then for a > ax,
G.1.1)
1 f
J _r,, n=i
For
/-, Z,
.7/1=1 71=1 71=1 7ll?=n
the series being absolutely convergent, and uniformly convergent in any
finite 2-range. Hence we may integrate term-by-term, and obtain
ambn2sm{T\ognlm)
3 "
n=l
2 r log n/m
The factor involving T is bounded for all T, m, and n, so that the double
series converges uniformly with respect to T; and each term tends to
zero as T -> oo. Hence the sum also tends to zero, and the result follows.

7.1 MEAN-VALUE THEOREMS 139
In particular, taking bn = dn and a = j3 = a, we obtain
rp
lim J- f |/(a+^)|2 dt = f 1^ (a > ax). G.1.2)
p
im J- f |/(a+^)|2 dt = f 1^ (a >
/TT 71 ~ 1
These theorems have immediate applications to ?(s) in the half-plane
a > 1. We deduce at once
^ f
-T
and generally
T
lim J_ f ^(a+it^iP-it) dt = ^+")(a+j3) (« > 1, /3 > 1). G.1.4)
Taking an = ^a(ti), we obtain
t TO
lim _L I* |?(a+*0|2* dt=y ^W- (CT > !)• C7-1-5)
J
rp
By A.2.10), the case k = 2 is
7'
f
J
-T
The following sections are mainly concerned with the attempt to
extend these formulae to values of a less than or equal to 1. The
attempt is successful for k < 2, only partially successful for k > 2.
7.2. We require the following lemmas.
Lemma. We have
VV i = 0(T2-2*logT) G.2.1)
Z, Z, mana\oQ,nm G
/or ^ < a < 1, arwi uniformly for \ < a < a0 < 1.
Let 1^ denote the sum of the terms for which m < \n, 22 the remain-
remainder. In EXJ logn/ra > -4, so that
m<n<T ^
22
m<n<T
In 22 we write m = n—r, where 1 < r < \n, and then
logn/m = — log(l—rjn) > r/n.
Hence
v, < A y y (n-r)^n-^ ^ y nl_2ff y 1 < ^T2_2ffl
Zw Z^ r/n ^w Z^ r

140 MEAN-VALUE THEOREMS Chap. VII
Lemma. V V _^ - = 0 S2-2 log ± . G.2.2)
0<m<n<oo G ' V '
Dividing up as before, we obtain
2i = 0[( t ^~ffe-s'1J] = O(S2"-2),
and f
n=1 r=1
Theorem 7.2. r
J
1
We have already accounted for the case a > 1, so that we now sup
pose that ? < a ^ 1. Since t'^ 1, Theorem 4.11, with x = t, gives
I
n<t
say. Now
c c
1 1
2V r (n\il , _
> m~lTn~a — at (T, = maxfm, n))
«i ^ T >i<^ V7 ..r » '
J
m<T
n<r ?-4 Z-i i log n/ra
== t v n-2ff- 2 ni-^+ol y y
2 + y y
provided that a < 1. If a = 1, we can replace the a of the last two terms
by f, say. In either case T
i
Hence
r
J i^E)i2^ - J \z\*dt+olj \z\t-°dt\+on t-*°dt\
i i • i ¦ ' \\ '
7'
- J
1
and the result follows.

7.2 MEAN-VALUE THEOREMS 141
It will be useful later to have a result of this type which holds
uniformly in the strip. It isf
Theorem 7.2 (A).
$ |?(a+ti)|2 dt < ^LTminflog T, -i^J
uniformly for \ < a < 2.
Suppose first that \ < a < |. Then we have, as before,
T
\Z\2dt<T
uniformly in a. Now
• < A log T
and also
7 A
< 1+ f u~2adu <
J a—-
Similarly T*~2° log T < T log T,
and also, putting x = Ba— l)log ^T,
This gives the result for a < f, the term 0(t~a) being dealt with as
before.
If § < a ^ 2, we obtain
T
f | Z|2 d? < T 2 n-t+0(!Tilog T),
and the result follows at once.
7.3. The particular case a = -| of the above theorem is
T
We can improve this O-result to an asymptotic equality 4 But Theorem
4.11 is not sufficient for this purpose, and we have to use the approxi-
approximate functional equation.
Theorem 7.3. As T->oo
T
0
Littlewood D). % Hardy and Littlewood B), D).

142 MEAN-VALUE THEOREMS Chap. VII
In the approximate functional equation D.12.4), take a — \, t > 2,
and x = 1/B^ log t), y = Vlogi. Then, since x(i+*0 = O{1),
*0 = 2 n-*-«(
^n<y
n<x
T
say. Since J (logfiJ dt = O(Tlog^T) = o (Tlog T),
2
it is, as in the proof of Theorem 7.2, sufficient to prove that
T
$ \Z\2dt~ TlogT.
0
T T
Now f \Z\2dt = f 2 w-i-a 2 n-$+ildt.
0 0
In inverting the order of integration and summation, it must be
remembered that a; is a function of t. The term in (m, n) occurs if
x > max(m,n) = T1/B7rVlogT1)
say, where T± = T^w.n). Hence, writing X = T/BWlog T),
T T
IZ |2 dt = J J f m-\-iln~\+u dt
2 2
m,n<X
yy_i_f
j
The first term is
TlogZ + 0(T) = TlogT+o {TlogT).
The second term is
Ol y Vlogn) —O(ZVlogZ) = 0(T),
and, by the first lemma of § 7.2, the last term is
0(ZlogZ) = O(TVlogT).
This proves the theorem.

7.4 MEAN-VALUE THEOREMS 143
7.4. We shall next obtain a more precise form of the above mean-
value formula, f
Theorem 7.4.
T
j \Ui+it)\2dt = TlogT+Br-l-log27r)T+O(Ti+«). G.4.1)
6
6
We first prove the following lemma.
Lemma. If n < T/2tt,
J {^^ ^Pl. G.4.2)
If n> T/27T, c > i,
c+iT
_,)„-, ds = O( ^ ) + 0(gL»). G.4.3)
We have
y(l-s) = 21-s7T-cosiS7rr(s) CT . ^i
A ' ^ v ' 2sin|s7rr(l — s)
This has poles at s = —2v (v = 0, 1,...) with residues
Also, by Stirling's formula, for — tt+8 < arg(—s) < tt—8
The calculus of residues therefore gives
i h -co+iT,
/ i-Wi h + iTi -co+
M S+ S+ J
— V
^ J-xT,
= 2 cos 27r?i = 2.
Also, since e(sa"«-s) = 0{e^'),
t Ingham A) obtained the error term 0(Tlog T); the method given here is due to
Atkinson A).

144 MEAN-VALUE THEOREMS Chap. VII
f n , J n\ ft ^ \4
x(l-s)n-° ds = 01 __-
, J J \l<T"r*-zil/
2 + iTi -oo
-i
= 0-.
nilog(T1/27ren)/'
and similarly for the integral over (—oo—iT^ \—iT1).
Again, for a fixed a,
(pj
Hence
s)n-s ds = n-\e-\™ J ei
$+;t t
where F{t) = flogt-«(log27r+l+logn),
F'(t) = Iogt—log2nn.
Hence by Lemma 4.2, the last integral is of the form
uniformly with respect to Tv Taking, for example, Tx — 2eT > Inert,
we obtain G.4.2). Again
c+iT T , T
(• f I t \c~\ / f
v(l—s)n-s ds = n-ce-*llT — e%F{-l) dt+Oln-0
c + t 1 1
and G.4.3) follows from Lemma 4.3.
In proving G.4.1) we may suppose that T\2-n is half an odd integer;
for a change of 0A) in T alters the left-hand side by 0(T?), since
t,(\-\-it) = 0(t\), and the leading terms on the right-hand side by
0(log T). Now the left-hand side is
T T
f i j r
J J
-T -T
= » J ^K(i-*)cto = i J
say

7.4 MEAN-VALUE THEOREMS 145
By G.4.2),
The first term isf
Since % d(n) = O(n€), the second term is
The last term is also clearly of this form. Hence
Ix =
Next, if c > 1,
f x(i-
^4 being the residue of 77^A— s)?2(s) at s = 1.
Since x(l—s) = 0(«ff-*), and ^2(a+*T) and J ^(n)n-s are both of
the form n<TI2iT
0(^-0+*) (a < 1), 0(T<) (a > 1),
the first term is O(Tz+e)-\-O(Tc-l+e).
By G.4.3), the second term is
d(n) I 1 w\\
2 shJ^M3*4 2 i
n>TI-n
Since c may be as near to 1 as we please, this proves the theorem.
A more precise form of the above argument shows that the error-
term in G.4.1) is O(TMog2T). But a more complicated argument,§
t See § 12.1, or Hardy and Wright, An Introduction to the Theory of Numbers, Theorem 320.
% Ibid. Theorem 315.
§ Titchmarsh A2).

146 MEAN-VALUE THEOREMS Chap. VII
depending on van der Corput's method, shows that it is O(Ti~*log2T);
and presumably further slight improvements could be made by the
methods of the later sections of Chapter V.
7.5. We now pass to the more difficult, but still manageable, case of
|?(s)|4. We first provef
Theorem 7.5.
T
liml f |
2'->» T J
i
Take x = y = *J{t/27r) and a > \ in the approximate functional equa-
equation. We obtain
2 ^ 2
n<-J(tl2n) n<V«/27r)
G 5 "H
say. Now
\mn)
where each variable runs over {1,^A/2^)}. Hence
T T
( \Z^dt=: f
J J
nixvY I
h
(mnixvY I \mn
where Tx — 2tt max(m2, ril, [x2, v-)
(mnJa '
The number of solutions of the equations mn = \xv = r is {d(r)}2 if
r < *]{TI2tt), and in any case does not exceed {d(r)}2. Hence
-t V
~ Z
r2'
Z
r<V(?/27r) V v(T/27r)
y&2 ^. G.5.2)
)
r=l
Hardy and Littlewood D).

7.5 MEAN-VALUE THEOREMS 147
Next V V <
f 4 \If lib I
(mnuv)a
and the right-hand side, by considerations of symmetry, is
Z, (mnixvY Zv mn
=
The remaining sum is
by the lemma of § 7.2. Hence
T
Now let j(T)= [\
i
T
i
s-l
nS
dt.
i
The calculations go as before, but with a replaced by 1 — a. The term
corresponding to G.5.2) is
T
r<AT
and the other two terms are 0{Ti+a+€) and 0(T2<T+€) respectively. Hence
and, since x(s) = 0(t?-a),
T T
T
-2) J fi-toftt) dt
J
The theorem now follows as in previous cases.
7.6. The problem of the mean value of | ?(!+*?) I4 is a little more
difficult. If we follow out the above argument, with a = ?, as accurately
as possible, we obtain
T
G.6.1)

148 MEAN-VALUE THEOREMS Chap. VII
but fail to obtain an asymptotic equality. It was proved by Ingham'j;
by means of the functional equation for (?(s)}2 that
G.6.2)
The relation f 1 ?($+#) I4 dt ~ Tl°g*T G.6.3)
J Ztt
l
is a consequence of a result obtained later in this chapter (Theorem 7.16).
7.7. We now pass to still higher powers of ?(s). In the general case
our knowledge is very incomplete, and we can state a mean-value
formula in a certain restricted range of values of a only.
Theoeem 7.7. For every positive integer k > 2
This can be proved by a straightforward extension of the argument of
§ 7.5. Starting again from G.5.1), we have
=y
(mv..mkn1...nk)<T \m1...mkj
where each variable runs over {1, ^/(t/2Tr)}. The leading term goes in the
same way as before, d(r) being replaced by dk(r). The main O-term is
of the form
\ ? ?> Jqr)°log r/qj
0<q<r<ATik
The corresponding term in
J(T) = J 2
T
2k
n
s-1
dt
n<V«/27r)
JL
is 0(Tk°+e),
and since \x\2k = 0(tk-2ka), we obtain 0(Tk{1-a)+€) again. These terms
are o(T) if a > l — l/k, and the theorem follows as before.
7.8. It is convenient to introduce at this point the following notation.
For each positive integer k and each a, let ^k{a) be the lower bound of
positive numbers | such that
Ingham A).

7.8 MEAN-VALUE THEOREMS 149
Each ju.fc(cr) has the same general properties as the function ju(ct)
denned in§ 5.1. By G.1.5), ^fc(a) = 0 for a > 1. Further, as a function
of a, ju.fc(cr) is continuous, non-increasing, and convex downwards. We
shall deduce this from a general theorem on mean-values of analytic
functions, f
Let f(s) be an analytic function of s, real for real s, regular for a ^ a
except possibly for a pole at s = s0, and O(ee[t{) as \t\ -> oo for every positive e
and a ^ a. Let a < /?, and suppose that for all T > 0
T
\ \f(<x+it)\* dt ^ C(T«+1), G.8.1)
o
T
J |/(j8+*0la* < C"(T» + 1), G.8.2)
o
where a > 0, b ^ 0, and C, C depend on f(s). Then for a < a < /?,
T>2, T
f |/(<r+^)|2 d« < K(CTa)^-^-a\C'Tb)^-^^\ G.8.3)
ir
where K depends on a, b, a, /? onZ?/, awci is bounded if these are bounded.
We may suppose in the proof that ol^ \, since otherwise we could
apply the argument to f{s-\-\—a). Suppose first that/(s) is regular for
a ^ <x. Let
G+100
If
I "TV o\ +/ oW*~S /j o (h(z\ (CT ^-¦> d' f\iV& % "CT* 77 1
2m J ^
G—X00
Putting z = ixe-i8 @ < 8 < \tt), we find that
are Mellin transforms. Let
Then, using Parseval's formula and Holder's inequality, we obtain
00
I(a) = 2tt j \(f>(ixe-i8)\2x2a-1 dx
27T
(a-o)/(j3-a)
t Hardy, Ingham, and Polya A), Titchmarsh B3).

150 MEAN-VALUE THEOREMS Chap. VII
Writing F(T) = J \f(<x+it)\2 dt < C(Ta+l)
o
we have by Stirling's theorem (with various values of K) '-
00
/(a) < K J («to-1
o
00
= K J
0
CO
KC
0
00
J
0
00
Similarly for 7(j3). Hence
/(a) <
Also 1/8 1/S
/(a) > X f [fia+iQlW-1 dt > iTS-2^1 [ |/(<r+
1/28 1./28
Putting 8 = 1/T, the result follows.
If f(s) has a pole of order k at s0> we argue similarly with (s—so)kf(s);
this merely introduces a factor T2k on each side of the result, so that
G.8.3) again follows.
Replacing T in G.8.3) by \T, \T,..., and adding, we obtain the result:
J \f(oc+it)\2dt = 0(T"), J |/(j3+»*)|*d* = 0{T%
T
then J |/(<7+iO|2 dt =
0
J
0
Taking/(s) = ^fc(s), the convexity of fxk(a) follows.

;9 MEAN-VALUE THEOREMS 151
.'.' ^ V 7.9. An alternative method of dealing with these problems is due to
.f His main result is
7.9. Let ak be the loiver bound of numbers a such that
t
± j \Z(*+it)\™ dt = 0A). G.9.1)
i
Then ak < max) 1 — —Z?--, i a j
/or 0 < a < 1.
We first prove the following lemma.
00
Lemma. Letf(s) = ]? <*nn~s be absolutely convergent for a > 1. Then
2 a & 1 f
_^e-8n = —. Y{iv—
c—i'oo
/or 8 > 0, c > 1, c > a.
For the right-hand side is
2tt* j ?-< nw ?-< ns 2tt% j
c — iao n — l n — \ c—ico
c — a + ioo
n— i *" ^
OO
s a
¦?—t n
e — a—i
00
n =
The inversion is justified by the convergence of
ne
Taking an = dk(n),f(s) = t,k(s), c = 2, we obtain
Zn = — [ T{w-s%k{w)hs-w dw [a < 2).
2tti j
i
ns j
2-ioo
Moving the contour to R(w) = <x, where a— 1 < <x < a, we pass the pole
of T{w—s) at w = s, with residue ?fc(s), and the pole of ?,k(w) at w = 1,
where the residue is a finite sum of terms of the form
t Carlson B), C).

152 MEAN-VALUE THEOREMS Chap. VII
This residue is therefore of the form 0{fr-1+H-JM), and, if 8 > \t\~A, it
is of the form O(e-Altl). Hence
Z-, ns 2tti J
a-too
Let us call the first two terms on the right Zx and Z2. Then, as in
previous proofs, if a > \,
I \Zl\* u = o(t y <%We-*A+o(y y
(
(
Z^ Z^ m<J-€na-€\iog:m/n\
by G.2.2). Also, putting w = oc+iv,
00
-a /*
J" J
¦r\ rrt
— 00
CO
V —00
f |T(m;—«)|d« [ \T(w—s)??k(w)\
The first integral is 0A), while for \t\ < T
-IT oo, ,-2T oo«
l
-IT oo, ,-2T oo«
J 4. j\\V(w-s)t>2k(w)\dv=l J + J \e-^"-^\v\Ak dv = 0(e~AT).
-00 IT' -00 IT'
J j J J
-00 IT' -00 IT'
Hence
r .it
J I |
^r ^ -it \t
IT ,
= 0 82*-2a I |^(a+i«)|2fcdt;
Hence
J |^E)|2fc^ =
Let 8 = 2T-iU+/*»<«»/U-a>, s0 that the last two terms are of the same order,
apart from e's. These terms are then 0(T) if
1—a

7.9 MEAN-VALUE THEOREMS 153
For such values of a, replacing T by \T, IT,..., and adding, it follows
that G.9.1) holds. Hence ak is less than any such a, and the theorem
follows.
A similar argument shows that, if we define a'k to be the lower bound
of numbers a such that
T
IJ \tta+it)\™ dt = O(T<), G.9.2)
l
then actually ak = ak. For clearly ak ^C ok; and the above argument
shows that, if a > ak, and a < a, then
T
f \Ua+it)\2kdt = O(T) + O(82ff-2-c)+O(82ff-2aT1+c).
iT
Taking S = T~x, where 0 < A < 1/B—2a), the right-hand side is O(T).
Hence ak ^ a, and so ak ^ a'k.
It is also easily seen that
{ n=l
For the term O(T) of the above argument is actually
, n n
n = 1 n = 1
and the result follows by obvious modifications of the argument. This
is a case of a general theorem on Dirichlet series.f
Theoeem 7.9 (A). // ju(ct) is the ^-function defined in § 5.1,
for k= 1,2,....
Since t(
J \?(a+it)\2k dt = 0lTW+e J \Z(a+it)\2k-2 dt ,
and hence
Since ^^(^fc-i) = 0, this gives i*.k(ok-x) < 2/Lt(afc_1), and the result
follows on taking a = ak_x in the previous theorem.
f See E. C. Titchmarsh, Theory of Functions, § 9.51.

154 MEAN-VALUE THEOREMS Chap. VII
These formulae may be used to give alternative proofs of Theorems
7.2, 7.5, and 7.7. It follows from the functional equation that
Since ju.fc(crfc) = 0, fxk(l—ak) ^ 0, it follows that ak ^ \. Hence, putting
a = 1— ak in Theorem 7.9, we obtain either ak = i or
i.e. 2ak-l ^ 2k(ak-
Hence a,. = ?, or
l<*(l-<rfc), afc<l-i. G.9.3)
For fc = 2 we obtain cr2 = |, but for k > 2 we must take the weaker
alternative G.9.2).
7.10. The following refinementf on the above results uses the
theorems of Chapter V on fx(a).
Theoeem 7.10. Let k be an integer greater than 1, and let v be determined
by 2+l < k < V2"-1+1. G.10.1)
Then ,,t<i__?H_. G.10.2)
The theorem is true for k = 2 (v = 1). We then suppose it true for
all I with 1 < I < k, and deduce it for k.
Take I = (v— lJ"-2+l, where v is determined by G.10.1). Then
oc) = 0, provided that
v
a> 2Z+2"-1-2 ~
Taking a = 1 — 2~v+1-\-e, we have, since
I f \Z(cc+it)\*k dt ^ max |^(a+^)|2fc-2^ f \?(*+it)\«dt,
T J K«^T 1 J
1
Davenport A), Haselgrove A).

7.10 MEAN-VALUE THEOREMS 155
by Theorem 5.8. Hence, by Theorem 7.9,
a, < 1 o
The theorem therefore follows by induction.
For example, if k = 3, then v = 2, and we obtain
instead of the result az ^ § given by Theorem 7.7.
7.11. For integral k, dk(n) denotes the number of decompositions of n
into k factors. If k is not an integer, we can define dk(n) as the coefficient
of n~s in the Dirichlet series for ?,k{s), which converges for a > 1.
We can now extend Theorem 7.7 to certain non-integral values of k.
THEOREMf 7.11. For 0 < k < 2
T
limi f \l{*+it)\»<dt = f^ (a > i). G.11.1)
This is the formula already proved for k = 1, & = 2; we now take
0 < k < 2. Let
The proof depends on showing (i) that the formula corresponding to
G.11.1) with ?N instead of ? is true; and (ii) that ?,N(s), though it does
not converge to ?(s) for a ^ 1, still approximates to it in a certain
average sense in this strip.
We have, if A > 0,
P<N
n
say, where the series on the right converges absolutely for a > 0, and
d\(n) = d\(ri) if n < N, and 0 ^ d\(ri) < d\(n) for all n. Hence
T
T-+00T
and T
1 "=1
lim lim -^
JV->oo T-+cc
1
im i f \ZN{*+it)\* dt = f {^- (a > i). G.11.3)
'—MX) J. I *—^ Iv
* n = l
f Ingham D); proof by Davenport A).

156 MEAN-VALUE THEOREMS Chap. VII
We shall next prove that
1 r
lim lim— |^(cr-f-i^) — ?N(a-\-it)\2kdt = 0 (a > ?). G.11.4)
iv->oo r->oo T J
i
By Holder's inequality
7'
Hi /
— \^\\ikK2~k) dt\
G.11.5)
Now {^^E) —I}2 is regular everywhere except for a pole at <s = 1, and
is of finite order in t. Also, for a > \,
T T
[ I1?iv(ct+*0 — l\*dt < ( {l + 2N\?(<j+it)\y dt = O(T).
J J
1 1
Hence, by a theorem of Carlson,f
T
T-
1 n =
for ct > \, where pN is the coefficient of n~s in the Dirichlet series of
{1n(s)—lY- ^ow Pi\(n) = 0 for n < iV, and 0 < pN{n) < ci(n) for all n.
Since 2 d2(n)n-2a converges, it follows that
T
lim linil f |^v(a+»0-l|4<tt - 0; G.11.6)
Ar->oo 2'->oo 1 J
1
G.11.4) now follows from G.11.5), G.11.6), and G.11.3).
We can now deduce G.11.1) from G.11.3) and G.11.4). We havej
(Jj {Jj
where J? = 1 if 0 < 2/c < 1, R = l/2k if 2k > 1. Similarly
(J
and G.11.1) clearly follows.
t See Titchmarsh, Theory of Functions, § 9.51.
J Hardy, Littlewood, and Polya, Inequalities, Theorem 28.

7.12 MEAN-VALUE THEOREMS 157
7.12. An alternative set of mean-value theorems.f Instead of
considering integrals of the form
T
0
where T is large, we shall now consider integrals of the form
00
J(8) = [ \?,(<j+it)\*ke-Bldt
6
where S is small.
The behaviour of these two integrals is very similar. If J(8) = 0A/8),
then r
I{T) < e j \Z{<j+it)\*ke-ilT dt < eJ{l/T) = 0(T).
o
Conversely, if I(T) = 0(T), then
00 00
J(S) = J I'(t)e-*dt = [J(Oe-*]o+8 J I(t)e~^dt
0 0
te-*fdt\ = 0A/8).
Similar results plainly hold with other powers of T, and with other
functions, such as powers of T multiplied by powers of log T.
We have also more precise results; for example, if I(T) ~ CT, then
J(S) ^ C/B, and conversely.
If I{T) ~ CT, let \J(T)-CT\ < eT for T > To. Then
To oo oo
J(B) = S [ I{t)e-lldt+B J {I{t)-Ct}e-Bldt+CB J ^e-s'^.
6 y« tv
The last term is Ce-8r»(To+l/S), and the modulus of the previous term
does not exceed e(T0+ 1/S). That J(B) ~ C/B plainly follows on choosing
first To and then S.
The converse deduction is the analogue for integrals of the well-known
Tauberian theorem of Hardy and Littlewood,J viz. that if an > 0, and
n=0 A—*
then 2 an ~ N.
n = 0
t Titchmarsh A), A9).
% See Titchmarsh, Theory of Functions, §§ 7.51-7.53.

158 MEAN-VALUE THEOREMS Chap. VII
The theorem for integrals is as follows:
#/@ ^ 0 for aM t> and
00
f/(Oe-*«fc~| G.12.1)
o
T
asS ->0, then J f(t) dt ~ T G.12.2)
o
as T -> oo.
We first show that, if P(x) is any polynomial,
I f(t)e-*lP(e-81) dt ~ I f P(z) cio;.
It is sufficient to prove this for P(x) = xk. In this case the left-hand
side is
00
0 0
Next, we deduce that
co 1
f/(Oe-*0(e-*) dt ~ If gr(ar) ^ G.12.3)
o o
if <7(a;) is continuous, or has a discontinuity of the first kind. For, given
e, we canf construct polynomials p(x), P(x), such that
p(x) < g(x) < P(x)
i i
and f {g(x)—p(x)} dx < e, | {P(x)—g(x)} dx < e.
0 0
Then
CO
limS f(t)e-8lq(e-81) dt < limS
o
1 1
= J P(x) dx < J 0(s)
o
and making e -> 0 we obtain
00 1
limS f f{t)e-*lg(e-*1) dt < f g(x) dx.
8-° i o
Similarly, arguing with p(x), we obtain
00 1
limS \ f{t)e-hlg{e-*1) dt > J g(x) dx,
8-»o 6 o
and G.12.3) follows.
f See Titchmarsh, Theory of Functions, § 7.53.

7.12 MEAN-VALUE THEOREMS 159
Now let
g(x) = 0 @ < x < e-1), = l(x (e-1 < x < 1).
oo 1/8
Then J f(t)e-*lg(e-81) dt = J f(t) dt
o o
i i
and ( g{x)dx= f — = 1.
0 l/e
1/8
r i
Hence I
o
which is equivalent to G.12.2).
^ 0 for all t, and, for a given positive m,
00
f(t)e~Bl dt ~ ^logm-, G.12.4)
J 6 5
o
jf(t)dt~ T\ogmT. G.12.5)
o
The proof is substantially the same. We have
00
1 , _( 1
]
dt
(k+l)BJ (Jfc+1)8
and the argument runs as before, with - replaced by -log771-.
5 So
We shall also use the following theorem:
00
// J/(Oe-8/<ft~ CS-a (a > 0), G.12.6)
i
00
then f t~$f{t)e-hl dt ~ cr^~^8^-« @ < B < «). G.12.7)
J "(a)
Multiplying G.12.6) by (S—^)p~1 and integrating over A7,00), we
obtain
00 00
J/@ dt J e-^S-i?)^-1 dS = C J

160 MEAN-VALUE THEOREMS Chap. VII
Now
J e-^S-T?)/3-1 dS = e-* J e-%^-1 dx = e-^
7, 0
00 00
and the remaining term is plainly o (?^~a) as rj -> 0. Hence the result.
7.13. We can approximate to integrals of the form J(S) by means of
Parseval's formula. If R(z) > 0, we have
2 + ioo x 2 + ioo
_L f T(sKk(s)z-s ds = y ^-1 f T(s)(nz)-Sds = fdk(n)e-n2,
2tti J 4-i 2rrl J »-i
2-ioo n -1 2-ioo
the inversion being justified by absolute convergence. Now move the
contour to a = a. @ < a < 1). Let Bk(z) be the residue at 5 = 1, so
that Bk(z) is of the form
Let <f>k(z)= fdk(n)e-~-Bk(z).
n = l
a + ioo
Then
1 f
—: I r(<s)? k(s)z~s ds = 4>j,(z). G.13.1)
a —ioo
Putting z = ia;e-iS, where 0 < S < \n, we see that
8)s G.13.2)
are Mellin transforms. Hence the Parseval formula gives
00
ixe-^x20-1 dx.
~ r \r(a+it)Ck(<r+it)\2eiir-^dt = f
0 G.13.3)
Now as \t\->co
\T(<j+it)\ = e-i^lt^-i^TT^l + Oit-1)}.
Hence the part of the ^-integral over (—oo, 0) is bounded as S -> 0, and
we obtain, for \ < a < 1,
J ^-i{i + O(r1)}|C(a+i0l2*e-28/^ = f Ifaiixe-^IW-1 dx+0(l).
0 "° G.13.4)

7.13 MEAN-VALUE THEOREMS 161
In the case a = ?, we have
|r(i+#)|2 = iraechirt = 27re-w»i+0(e-3wi'i).
The integral over (—oo, 0), and the contribution of the 0-term to the
whole integral, are now bounded, and in fact are analytic functions of S,
regular for sufficiently small |S|. Hence we have
00 00
J \Ui+it)\*ke-*tdt = \ \4>k(ixe-*)\*dx+0{l). G.13.5)
7.14. We now apply the above formulae to prove
Theorem 7.14. As S -> 0
00
J
~ ^log^
^^ G.14.1)
In this case Bx(z) = 1/z, and
Hence G.13.5) gives
00
-2Sldt =
f
dx+O{l).
0 ° G.14.2)
The a;-integrand is bounded uniformly in S over @, rr), so that this part
of the integral is 0A). The remainder is
f
J
\exp(—zo;el8) —
77
00
J (JbjC • • 5 j A (LOC
I I 1?^Q I
¦ i O"v"r^ i ^ 'y*^ — *° i I ' / ovn i — i ^vo *° i Is. i pyt^i ^ 'V^ *° i I 'V
77 7T
00 00
_ie-i8 r ^j^ ^+ r^. G>14^
77
The last term is a constant. In the second term, turn the line of integra-
integration round to (rr, tt+zoo). The integrand is then regular on the contour
for sufficiently small |S|, and is 0{a;-1exp(— a;cos8)} as x -> oo. This
integral is therefore bounded; and similarly so is the third term.

162 MEAN-VALUE THEOREMS Chap. VII
The first term is
GO
JCO «J^
y T exp(—imxe~l°-{-inxe™) dx
in S—z(m—n)rr cos i
77
co
(m-\-n)sm$-\-i(m—tj)cosS
x C (™+7i)sinScos{(ra—7i)-rrcosS}^_(,.
n==l ?/i = 2 n = l
co in —
_ 2 V V (m-^)cos s sin{(m-7tOr cos 8} e_(m+nOr sin 8
Z ^ (+Ji28 + (—7iJcos2S
i=2 n = l
the series of imaginary parts vanishing identically. Now
¦ ^ 2 > > / ""'72"" 2Se-m7rsin8 = 0(8 2
Z-< Z-i (m —7iJCOS2S \ m =
771=2 71 = 1 V '
and, since |sin{(m—7iOrcosS}| = |sin{2(m—wOrsm2|S}| = 0{(m—w)S2},
(CO 771 — 1 \ / co ,
Ol > > e-»W77&ln<5 = (J [d? > mg-77l77Sin6 —- (J[i.).
77l=*2n=l 771=2 '
This proves the theorem.
The case \ < a < 1 can be dealt with in a similar way. The leading
term is
CO CO
3.2G-1
X />-2«xsinSr2a-l fjr
77
CO
BsinS)'2<' J e'—l BSJ" J e»—1
i8
J () J
27rsin8 0
CO
siu 8+i(/7i-n)cos,
Also (turning the line of integration through — \tt)
CO
J
CO
8
0
? , G.14.4)
m—n /

¦f
7.14 MEAN-VALUE THEOREMS 163
and the terms with m ^n give
ol
(
oo
Hence J ^-i| ?(a+*0 |2e~28' <fc ~ ' ^g^"' • G.14.5)
o
Hence by G.12.6), G.12.7)
'2"-. G.14.6)
J
7.15. We shall now show that we can approximate to the integral
G.14.1) by an asymptotic series in positive powers of S.
We first requiref
Theorem 7.15. As z->0 in any angle |argz| < A, where A < \n,
f d(n)e— = Yll^l_|_I_|_ N^bnz***+O{\z\*»), G.15.1)
n=l 2 4 n = 0
6n are constants.
n
Near s = 1
— |_ i o_ 1_
2E-lJ"t" 2 5_ 1 "•"-•
Hence by G.13.1), with A = 2,
f
@ < « < 1).
a —too
Here we can move the line of integration to a = —2N, since
T(s) = 0{\t\Ke~\^), ?2(<s) = O(\t\K) and z~s = O(r~ae^). The residue
at 5 = 0 is ?2@) = J. The poles of T(s) at 5 = — 2n are cancelled by
zeros of ?2E). The poles of T(s) at 5 = — 2n— 1 give residues
The remaining integral is O(|z|2iV), and the result follows.
The constant implied in the 0, of course, depends on N, and the
series taken to infinity is divergent, since the function ^ d{n) e~nz
cannot be continued analytically across the imaginary axis.
f Wigert A).

164 MEAN-VALUE THEOREMS Chap. VII
We can now provef
Theorem 7.15 (A). As S -+ 0, for every positive N,
N
I
n = 0
00
J
2sinS ^ Z n
o
the constant of the 0 depending on N, and the cn being constants.
We observe that the term 0A) in G.14.2) is
oo
rttdt-\ J \m+it)\2e^-2^sec\i7Tt dt,
0 —oo
and is thus an analytic function of S, regular for |S| < n. Also
77
J
,W
_ ixe~i8) — 1 ixe~i8j (exp(—ixeiS)— 1 ixeio)
is analytic for sufficiently small | S |. We dissect the remainder of the
integral on the right of G.14.2) as in G.14.3). As before
77+100
r i dx __ r
J exp(—ixeiZ) — 1 x J
dz
exp(—izeih) —
and the integrand is regular on the new line of integration for sufficiently
small 181, and, if S = ?-\-ir), z = n-\-iy, it is 0{2/~1exp(—ycostje'7!)} as
?/ -> oo. The integral is therefore regular for sufficiently small | S |.
Similarly for the third term on the right of G.14.3); and the fourth
term is a constant.
By the calculus of residues, the first term is equal to
?* ie~iS exp(—2imre2i8) —
[p{(i7T—y)e-ih}— l][exp{(—iir-\-y)ei8}—
As before, the ^/-integral is an analytic function of S, regular for |S|
small enough. Expressing the series as a power series in expBz7re2iS),
we therefore obtain
f I C(H*0l2e-28< dt = 27reiS f d(n)ex^Binne^)+ f anSn G.15.2)
g n=l n=0
for ISI small enough and R(S) > 0.
f Kober D), Atkinson A).

7.15 MEAN-VALUE THEOREMS 165
Let z = 2i7r(l-e2i8) in G.15.1). Multiplying by 2nei?>, we obtain
is ^r
- ovxx y—logD7relSsin8) 1
.
,
n = 0
and the result now easily follows.
7.16. The next case is that of |?(?+^)|4.
In G.14.2) the contribution of the a:-integral for small x was negligible.
We now take G.13.5) with k = 2, and
U*) = I d(n)e-™-7-—^. G.16.1)
n=l Z
In this case the contribution of small x is not negligible, but is sub-
substantially the same as that of the other part. We have
a+ioo
= 2^ J
a —too
1—a + ioo
J
s)z1-s ds
1 —a—ioo
1 —a+ioo
J
1—a—ioo
Now
|sin?s7r|/j
If z = ixe~i8 (x > x0, 0 < 8 < \n), the 0 term is
1—a + ioo
01
-> ±00).
1- J
1 —a—io
-loo
uniformly for small 8. Hence
1 —a+ioo
^/I\ = ZL^ f 21-2s7T1-2sr{sK2{s)z~s ds+O{x«)
\z) 2m J
1 —a—ioo
= -2Trizcf>2{4Tr2z) + O{x«), G.16.2)
where a may be as near zero as we please.

166 MEAN-VALUE THEOREMS Chap. VII
We also use the results
f d\n)e-^ = ol- log3 -), G.16.3)
n=i \v -nl
f nU\n)e~^ = o(i-log3i) G.16.4)
as 7y->0. By A.2.10)
2+ioo ^ 2 + ioo
2-too
oo
2-ioo V ' »-l
= 2 d2{n)e-nri. G.16.5)
Hence
c + ioo
^7^^ J
C —loo
where
where
B
a,
is
b,
the residue at 5 =
R_ 1/ b 3
~~ 7] \ ^
c, d are constants,
/7. -
1; and
1
\-0 10]
V
and in
1
21
fact
1
1
This proves G.16.3); and G.16.4) can be proved similarly by first
differentiating G.16.5) twice with respect to -q.
We can now provef
Theorem 7.16. As 8 -> 0
J
o
Using G.13.5), we have
J
o
and it is sufficient to prove that
2tt
f Titchmarsh A).

f
7.16 MEAN-VALUE THEOREMS
For then, by G.16.2),
2tt
f I <f>2(ixe-*) \>dx= f <f>2 l^f] 2 ^ = f
0 1/27T 1/27T
oo
-;
1/27T
;-)|Ǥ @
a;51
1/27T
oo
2tt
oo
= j \<f>2{ixe-i8)\2dx+0l j \<f>z(ize-i8)\2 dx j
2n ^2tt 2tt
J x2a-2
and the result clearly follows.
It is then sufficient to prove that
r
for the remainder of G.16.1) will then contribute O(S-Mog2l/8).
As in the previous proof, the left-hand side is equal to
o —imr sin 8
oo
n =
' 27isinS '
oo m-1
+ 2 y y d{m)d{n)
167
\ V / / p— 2Gn+n)ir sin 8
oo m — 1
9
(w+7iJsin28+(m—nJ cos28
nN (m-n)cos 8 sin{2(m-nO7 cos 8} 2{m+n>n8in 8
= 2 n =
Now
] — __t_(l e-47TSin8\ X e-47rnsin8
1 2 sin 8 Z/ ^_ v
n = l
n
oo
277
n
L f
-47rxsin
oo
If I V \ 1 1
= o 8 . g e-vlog4 y. J^y^F-2glog4s'
8ttz sin 8 J \47r sin o) ott^o o

168 MEAN-VALUE THEOREMS Chap. VII
00
IS. I <2
{m—nJcos28
m d(m) e~
cos28 ^ v ' Z, r2
m = 2 r=l
4* r\
sin 8
00
— ^ m d(m)d(m—r)e~
cos28
r=l
The square of the inner sum does not exceed
m2d2{m)e-2™8in8 f d2(m—i
m=r+l
d2{m)e-2™8in s
by G.16.3) and G.16.4). Hence
Finally (as in the previous proof)
This proves the theorem.
It has been proved by Atkinson B) that
00
j )\*e-* dt
wnere .A — — 5, x> — ri-i.vy6-,, "^-1 —
A method is also indicated by which the index jf could be reduced to §.
7.17. The method of residues used in § 7.15 for | ?($+?$) I2 suggests
still another method of dealing with | ?(?+*?) I4* This is primarily a
question of approximating to
f °° 2 f
V dGi)exp(—inxe~i8) dx = \
2tt
oo oo
= 22/
m = l n=l o*l
dx
i8) — l}{exp(—inxei8)
27T

7.17 MEAN-VALUE THEOREMS 169
In the terms with n ^ m, put x = ?/ra. We get
yly f
??im n=m Jm
[i? e~iS) — l}{exp(—ii
2nm
Approximating to the integral by a sum obtained from the residues of
the first factor, as in § 7.15, we obtain as an approximation to this
1
exp{—2i{nr/mOre2i8}—'.
oo oo
oo , oo oo
/ oo - oo oo
i
1—expBivm-17re2iS)|/'
The terms with m I v are
m\v
The remaining terms are
oo
( 1
m ^ ?-i l/m
m = l fc = l Z = l '
/ oo oo
= 0 y y e
Z^
rn = lk = l Z = l
/ oo oo km.
(
m=lfc=l
Z=l
^Z=l

170 MEAN-VALUE THEOREMS Chap. VII
Using Schwarz's inequality and G.16.3) we obtain
1\ ^/:
Actually it follows from a theorem of Ingham A) that this term is
7.18. There are formulae similar to those of § 7.16 for larger values
of k, though in the higher cases they fail to give the desired mean-value
formula, f
We have
a+ioo
J
a —too
1-a + ioo
—s)tk(l—s)z1-»ds
1 —a —too
1 — a+ ioo
__L f
2-rri J
Xk(s)
l — oc — ioo
\stt cosec 7TsTk-1{s).
For large 5 F
Now r{(ifc—l)a—;
Hence we may expect to be able to replace FA:~1E) by
Also, in the upper half-plane,
*¦" cosec 577 ~ (ie-i*'*w)fc —^ = —21-kie-isn{*k-1).
We should thus replace F(l— s)x~k{s) by
—i. 21-k*7T1-kse-is*1*k-1)(k—l)-<*-1>»+-J*-JB7r)-
Hence an approximation to <f>k(ljz) should be
71=1
1 — a + too
-«Bfc7rfr7i2)-s ds.
J r{(ifc—1M—
X
1— a—ioo
•j- See also Bellman C).

7.18 MEAN-VALUE THEOREMS 171
Putting s = {w+\k— l)/(k— 1), the integral is
2ttij
= —i{27T)$kz{k— lySe-iMik-imk-D^Wn^-dk-mk-i) x
X exp{- (k— i)e^(^
Putting z = ia;e~iS, we obtain
B7r)fc/B*-2)(ifc— l)—^/{^-^-(AA-iMfc-i) x
X Ck exp{—(jfc—1 )ei
where \Ck\ = 1.
We have, by G.13.5),
CO W
j \m+it)\2ke-2*ldt = J \<f>k{ixe-i8)\2dx+O{l)
o o
A oo
= j \<f>k(ixe-i8)\2 dx+ J|^
o A
As in the above cases, the integral over (A, oo) is
VrZ M<Z fa^ ins cos{A(m-n)cos8} A(OT+n)8in
Z dk{m>dk{n> (m+nJsm28 + (m-nJ cos^S *
- 2-V "V dJwJdJn) (^-^)cosSsin{A(m-n)cos8}e_A(OT+n)9in
Z 2L, kK ] kK ] (m4-7iJsin28+(m-7iJcos2S
w-2 n=l V i / \ /
3. G.18.1)
Using the relation dk{n) = 0(n€), we obtain
and, since (mH-7iJsin28+(m—?iJcos28 > Ah{m-\-n){m—n),
7/1—1 i \
V _L_ = Of f me
n = 1
(
y m«e^msin8 V _L_ = Of f mee-^sinS) = 0 -
Zv Z / \^ 7 \(AS
m = 2
(
y me _L_ = Of f me) = 0 ^
@0 7ft —1
y meAis y = o
Hence, for A < A,

172
MEAN-VALUE THEOREMS
Chap. VII
Also
J
l/A
2dx
X2
and by the above formula this should be approximately
X
ifc-l
oo
f
l/A
Putting a; = I*-1, this is
a;2-fc/(fc-l)'
oo
,,,_., n=l
X
and we can integrate as before. We obtain
dk{m)dk{n) _ ^
771=1 71 = ]
exp[(&—
cos S/(Jfc— 1) —
x
where i? depends on A: only.
The terms with m = n are
{\ f
— 1)—
in S/(Jfc—
The rest are
1 exp( —
Now
TO-1
777-1
2.
n=l

7.18 MEAN-VALUE THEOREMS 173
Hence we obtain
0{
771=2
" 0
1/**-1))} = o\ Jxeex^{-
Altogether
CO
1
0
and taking A = S^-1, we obtain
oo
\^(^-{-it)\2ke~28t dt = 0(8~ik~€) (k ^ 2).
o
This index is what we should obtain from the approximate functional
equation.
7.19. The attempt to obtain a non-trivial upper bound for
!)l2fce-*efc
o
for k > 2 fails. But we can obtain a lower boundf for it which may be
somewhere near the truth; for in this problem we can ignore <f>k(ixe-i8)
for small x, since by G.13.5)
oo oo
J \t{$+it)\2ke~281 dt > J \<f>k(ixe-i8)\2dx+O{l), G.19.1)
o i
and we can approximate to the right-hand side by the method already
used.
If k is any positive integer, and a > 1,
j ]\~k = T-T Y (fc + Wl—1)! J_ = V d*(w)
«s/ 1 1 Z, (jfc—l)!m! vms
p v r ' p w=0 ' - n=l
If we replace the coefficient of each term p~ms by its square, the
coefficient of each n~s is replaced by its square. Hence if
71"
71 = 1
then FJs) =
P 771=0
t Titchmarsh D).

174 MEAN-VALUE THEOREMS Chap. VII
say. Thus A(i) = 1 +| + ...,
and
H)H-f+- -M*-)- Hi)
Hence the product
H)*4
HLa
is absolutely convergent for a > -|, and so represents an analytic
function, g(s) say, regular for a > \, and bounded in any half-plane
a > 1+8; and' j, (<) =
f (Z|Gi)e-2w8inS = -L f
n = l 2zr* J
Now
n = :
2—tco
Moving the line of integration just to the left of a = 1, and evaluating
the residue at 5 = 1, we obtain in the usual way
6
Similarly
2+ too
2^ f
n=l J.
2 — 100
since here there is a pole of order k2 +1 at 5 = 0.
We can now prove
Theorem 7.19. For any fixed integer k, and 0 < S < So = B0(k),
00
J
J
0
The integral on the right of G.19.1) is equal to G.18.1) with A = 1;
and n
while S
The result therefore follows.

7.20
MEAN-VALUE THEOREMS
NOTES FOR CHAPTER 7
175
7.20. When applied (with care) to a general Dirichlet polynomial, the
proof of the first lemma of § 7.2 leads to
N
N
y an-"
La n
1
However Montgomery and Vaughan [1 ] have given a superior result,
namely
N
a n~u
n
N
G.20.1)
Ramachandra [2] has given an alternative proof of this result. Both
proofs are more complicated than the argument leading to G.2.1).
However G.20.1) has the advantage of dealing with the mean value of
C(s) uniformly for a ^ \. Suppose for example that a = \. One takes
x = IT in Theorem 4.11, whence
say, for T ^ t ^ 2T. Then
IT
\\Z\2dt=
J
Moreover Z<^T%, whence
Then, since
we conclude that
IT
1
[ \?$ + it)\2dt = T\ogT+O(T),

176 MEAN-VALUE THEOREMS Chap. VII
and Theorem 7.3 follows (with error term O(T)) on summing over
\ T, \ T,... .In particular we see that Theorem 4.11 is sufficient for this
purpose, contrary to Titchmarsh's remark at the beginning of §7.3.
We now write
Much further work has been done concerning the error term E(T). It
has been shown by Balasubramanian [1] that E{T) <^ T$ + e. A different
proof was given by Heath-Brown [4]. The estimate may be improved
slightly by using exponential sums, and Ivic [3; Corollary 15.4] has
sketched the argument leading to the exponent -j^- + e, using a lemma
due to Kolesnik [4]. It is no coincidence that this is twice the exponent
occurring in Kolesnik's estimate for /zQ), since one has the following
result.
Lemma 7.20. Let k be a fixed positive integer and let t ^ 2. Then
log2*
tt\ + it)k < (logt)(l+ I \C^ + it + iu)\ke-^du\ G.20.2)
-log2*
This is a trivial generalization of Lemma 3 of Heath-Brown [2], which
is the case k = 2. It follows that
C(? + itJ < (logt)* + (logt)maxE{t±(logtJ}. G.20.3)
Thus, if n is the infimum of those a for which E(T) <Ta, then /*(J) ^ \\i.
On the other hand, an examination of the initial stages of the process
for estimating (,(\ + if) by van der Corput's method shows that one
is, in effect, bounding the mean square of C(^ + it) over a short range
(t — A, t + A). Thus it appears that one can hope for nothing better for
/z(^), by this method, than is given by G.20.3).
The connection between estimates for CQ + it) and those for E(T)
should not be pushed too far however, for Good [1 ] has shown that
E(T) = Q(T$. Indeed Heath-Brown [1] later gave the asymptotic
formula
T
E{tJdt = ^Bn)-^^-T2 + O(Thog2T) G.20.4)
CC)
o
from which the above Q-result is immediate. It is perhaps of interest to

7.20 MEAN-VALUE THEOREMS 177
note that the error term of G.20.4) is Q(T*(log T)~*). This has been
proved by Meurman in unpublished work, using an argument analogous
to that employed in the proof of Lemma a in §14.13. He has also re-
reduced the error term in G.20.4) to O(T(logTM), so as to include
Balasubramanian's bound E(T)<T$ + e.
Higher mean-values of E(T) have been investigated by Ivic [1] who
showed, for example, that
T
E(t)8dt<$T3+E. G.20.5)
This readily implies the estimate E(T) <
The mean-value theorems of Heath-Brown and Ivic depend on a
remarkable formula for E(T) due to Atkinson [1]. Let 0 < A < A' be
constants and suppose AT ^ N ^ A' T. Put
T N /NT N2
2n 2 \2n 4
Then E(T) = ?x + ?2 + O(log2 T), where
G.20.6)
with
f(n) = ^-7r + 2Tsinh-1(^M +(n2n2 + 2nnT)K G.20.7)
and
Z2 = 2 X d(n)n~H log-—) smg(n)
where
Atkinson loses a minus sign on [1; p 375]. This is corrected above. In
applications of the above formula one can usually show that Z2 may be
ignored. On the Lindelof hypothesis, for example, one has

178 MEAN-VALUE THEOREMS Chap. VII
for x ^ T, so that Z2 < T& by partial summation; and in general one finds
Z2 < T2^ + e. The sum Z1 is closely analogous to that occuring in the
explicit formula A2.4.4) for A(x) in Dirichlet's divisor problem. Indeed, if
n = o(T § then the summands of G.20.6) are
7.21. Ingham's result has been improved by Heath-Brown [4] to give
T
[\?$ + it)\*dt = ? cnT(log T)» + O(Tl+E) G.21.1)
J n = 0
0
where c4 = Bk2)-1 and
c3 = 2{4y-l-logB7r)-12C'BOr-2}7r-2.
The proof requires an asymptotic formula for
d(n)d(n + r)
with a good error term, uniform in r. Such estimates are obtained in
Heath-Brown [4] by applying Weil's bound for the Kloosterman sum
(see §7.24).
7.22. Better estimates for ak are now available. In particular we have
ff3 ^ ty an(^ ff4 ^ f- The result on aA is due to Heath-Brown [8]. To
deduce the estimate for a3 one merely uses Gabriel's convexity theorem
(see § 9.19), taking a = \, p = |, X = \, n = \, and a = ^.
The key ingredient required to obtain c4 ^ ^ is the estimate
T
[ \a\ + it)\12dt<T2 (log TI7 G.22.1)
o
of Heath-Brown [2]. According to G.20.2) this implies the bound
) ^ i. In fact, in establishing G.22.1) it is shown that, if \t$ + itr)\
V (> 0) for 1 ^ r ^ R, where 0 < tr ^ T and tr + 1-tr^l, then
and, if V ^ T *(log T7J, then
R< TV-6(\ogT)8.
Thus one sees not only that C,{\ + it) ^ tf (log t)$, but also that the number

7.22 MEAN-VALUE THEOREMS 179
of points at which this bound is close to being attained is very small.
Moreover, for V ^ T&(log TJ, the behaviour corresponds to the, as
yet unproven, estimate
T
I
To prove G.22.1) one uses Atkinson's formula for E(T) (see §7.20) to
show that
T+G
T-G
K
K
[ \S(x)\dx)e-G2K'T, G.22.2)
J /
where K runs over powers of 2 in the range T$ ^ K ^ TG~ 2log3T, and
S(x) = S(x, K,T)= X (-1)" dW
with f(n) as in G.20.7). The bound G.22.2) holds uniformly for log2 T^G
^ T&. In order to obtain the estimate G.22.1) one proceeds to estimate
how often the sum S(x, K,T) can be large, for varying T. This is done by
using a variant of Halasz's method, as described in §9.28.
By following similar ideas, Graham, in work in the process of
publication, has obtained
T14(logTL25. G.22.3)
Of course there is no analogue of Atkinson's formula available here, and
so the proof is considerably more involved. The result G.22.3) contains
the estimate /z(f) ^ ^ (which is the case I = 4 of Theorem 5.14) in the
same way that G.22.1) implies /z(^) ^ ?.
7.23. As in §7.9, one may define ak, for all positive real k, as the
infimum of those a for which G.9.1) holds, and a'k similarly, for G.9.2).

180 MEAN-VALUE THEOREMS Chap. VII
Then it is still true that ak = a'k, and that
T
I
<r + H)\2k dt = TYJ dk(nJn-2a + O(T1~d)
k
1
1
for a > ak, where S = {a, k) > 0 may be explicitly determined. This may
be proved by the method of Haselgrove [1]; see also Turganaliev [1]. In
particular one may take S(a, J) = \{a — J) for \ < a < 1 (Ivic [3; (8.111) ]
or Turganaliev [1]). For some quite general approaches to these
fractional moments the reader should consult Ingham D) and Bohr and
Jessen D).
Mean values for a = \ are far more difficult, and in no case other than
k = 1 or 2 is an asymptotic formula for
T
I
say, known, even assuming the Riemann hypothesis. However Heath-
Brown [7] has shown that
T(log T)k2<Ik{T)<T(log T)
k2
Ramachandra [3], [4] having previously dealt with the case k =
Jutila [4] observed that the implied constants may be taken to be
independent of k. We also have
Ik{T)>T{\ogT)k2
for any positive rational k. This is due to Ramachandra [4] when k is
half an integer, and to Heath-Brown [7] in the remaining cases.
(Titchmarsh [1; Theorem 29] states such a result for positive integral k,
but the reference given there seems to yield only Theorem 7.19, which is
weaker.) When k is irrational the best result known is Ramachandra's
estimate [5]
Ik(T)> T(logT)k2(\og\ogT)-k2.
If one assumes the Riemann hypothesis one can obtain the better results
Ik(T)<T (log T)k2 @<&<2)
and
Ik(T)>T(\ogT)k2 (k^O), G.23.1)

7.23 MEAN-VALUE THEOREMS 181
for which see Ramachandra [4] or Heath-Brown [7 ]. Conrey and Ghosh
[1] have given a particularly simple proof of G.23.1) in the form
I
with
They suggest that this relation may even hold with equality (as it does
when k = 1 or 2).
7.24. The work of Atkinson B) alluded to at the end of §7.16 is of
special historical interest, since it contains the first occurence of
Kloosterman sums in the subject. These sums are defined by
S(q;a,b) = ? exp f — (an + bh) ), G.24.1)
n= 1
n. q) = 1
where nh = l(mod q). Such sums have been of great importance in
recent work, notably that of Heath-Brown [4] mentioned in §7.21, and of
Iwaniec [1] and Deshouillers and Iwaniec [2], [3] referred to later in
this section. The key fact about these sums is the estimate
\S(q; a, b)\ ^ d(q)qHq, a, b)K G.24.2)
which indicates a very considerable amount of cancellation in G.24.1).
This result is due to Weil [1 ] when q is prime (the most important case)
and to Estermann [2 ] in general. Weil's proof uses deep methods from
algebraic geometry. It is possible to obtain further cancellations by
averaging S(q; a, b) over q, a and 6. In order to do this one employs the
theory of non-holomorphic modular forms, as in the work of
Deshouillers and Iwaniec [1 ]. This is perhaps the most profound area of
current research in the subject.
One way to see how Kloosterman sums arise is to use G.15.2). Suppose
for example one considers
L u-u
e-t/Tdt. G.24.3)
0
Applying G.15.2) with 26 = 1/T + i \og(v/u) one is led to examine
Bninu
d()\l/
n=

182 MEAN-VALUE THEOREMS Chap. VII
One may now replace ei/Tby 1 + (?/ T) with negligible error, producing
f Tv
2-joo
where
u\ _ ^ /2ninu\
U/ n=l
This Dirichlet series was investigated by Estermann [1], using the
function C(s, a) of §2.17. It has an analytic continuation to the whole
complex plane, and satisfies the functional equation
providing that (u, v) = 1. To evaluate our original integral G.24.3) it is
necessary to average over u and v, so that one is led to consider
=\ u^U \ U /
(u» v) = 1 (u. u) = 1
for example. In order to get a sharp bound for the innermost sum on the
right one introduces the Kloosterman sum:
' 1 u = m(mod d)
27iia(m —w)'
m=l \ " / !*sSC7
(u. u) = 1 ' (m« u) = 1 u = m(modu)
=.?, E) ^
(m. v) = 1
= - Z S(v;a,n) 2, exp
V a= 1 «<t/ V U /
and one can now get a significant saving by using G.24.2). Notice also
that S(v; a, n) is averaged over v, a and n, so that estimates for averages
of Kloosterman sums are potentially applicable.
By pursuing such ideas and exploiting the connection with non-
holomorphic modular forms, Iwaniec [1] showed that
Z I IC(i +

I
7.24 MEAN-VALUE THEOREMS 183
for 0 ^ tr ^ T, tr+1-tr ^ A ^ Tk In particular, taking R = 1, one has
G.24.4)
which again implies /z(?) ^ ?, by G.20.2). Moreover, by a suitable choice
of the points tr one can deduce G.22.1), with T2+e on the right.
Mean-value theorems involving general Dirichlet polynomials and
partial sums of the zeta function are of interest, particularly in
connection with the problems considered in Chapters 9 and 10. Such
results may be proved by the methods of this chapter, but sharper
estimates can be obtained by using Kloosterman sums and their
connection with modular forms. Thus Deshouillers and Iwaniec [2 ], [3 ]
established the bounds >
T
I
Kl2
G.24.5)
and
T
I W11
dt
am\A( ? \bn\A G.24.6)
/ \nsZN /
for N ^ M. In a similar vein Balasubramanian, Conrey, and Heath-
Brown [1] showed that
T
*dt = CT+OA{T(log T)-A},
G.24.7)
C=
m, re ^ M
mn
T(m,nJ
2nmn
2y-l
for M ^ T & e, where A is any positive constant, and the function F
satisfies F(x) <$ 1, F'(x) <? x~1. The proof requires Weil's estimate for the
Kloosterman sum, i

VIII
^-THEOREMS
8.1. Introduction. The previous chapters have been largely con-
concerned with what we may call 0-theorems, i.e. results of the form
M = 0{f(t)}, l/?(«) = 0{g(t)}
for certain values of a.
In this chapter we prove a corresponding set of ?3-theorems, i.e. results
of the form m
the Q, symbol being defined as the negation of o, so that F(t) = Q.{<f>(t)}
means that the inequality \F(t)\ > A<f>(t) is satisfied for some arbitrarily
large values of t.
If, for a given function F(t), we have both
F(t) = 0{f(t)}, F(t) = Q{f(t)},
we may say that the order of F(t) is determined, and the only remaining
question is that of the actual constants involved.
For a > 1, the problems of t,(o-\-it) and \jl{a-\-it) are both solved.
For \ ^ a ^ 1 there remains a considerable gap between the 0 -results
of Chapters V-VI and the ^-results of the present chapter. We shall
see later that, on the Riemann hypothesis, it is the Q.-results which
represent the real truth, and the 0-results which fall short of it. We
are always more successful with ^-theorems. This is perhaps not
surprising, since an O-result is a statement about all large values of t,
an ^-result about some indefinitely large values only.
8.2. The first Q. results were obtained by means of Diophantine
approximation, i.e. the approximate solution in integers of given equa-
equations. The following two theorems are used.
Dibichlet's Theorem. Given N real numbers a1} a2,..., aN, a positive
integer q, and a positive number tQ, we can find a number t in the range
to<t^toq», (8.2.1)
and integers xx, x2,..., xN, such that
\tan-xn\ < l/q (n = 1, 2,..., N). (8.2.2)
The proof is based on an argument which was introduced and employed
extensively by Dirichlet. This argument, in its simplest form, is that,
if there are m-\-\ points in m regions, there must be at least one region
which contains at least two points.

8.2 Q-THEOREMS 185
Consider the 2V-dimensional unit cube with a vertex at the origin and
edges along the coordinate axes. Divide each edge into q equal parts,
and thus the cube into qN equal compartments. Consider the qN-\-l
points, in the cube, congruent (modi) to the points (uavua2,...,uaN),
where u = 0, t0, 2t0,..., qNt0. At least two of these points must lie in the
same compartment. If these two points correspond to u = uv u = u2
(ux < u2), then t = u2—ux clearly satisfies the requirements of the
theorem.
The theorem may be extended as follows. Suppose that we give u
the values 0, tQ, 2tQ,..., mqNtQ. We obtain mqN-\-l points, of which one
compartment must contain at least m-\-l. Let these points correspond
to u = uv..., um+v Then t = u2—u1,..., um—u1, all satisfy the require-
requirements of the theorem.
We conclude that the interval (tQ, mqNtQ) contains at least m solutions
of the inequalities (8.2.2), any two solutions differing by at least t0.
8.3. Kronecker's Theorem. Let av a2,..., aN be linearly independent
real numbers, i.e. numbers such that there is no linear relation
X1a1-\-...+XNaN = 0
in which the coefficients Xv... are integers not all zero. Let bv..., bN be any
real numbers, and q a given positive number. Then we can find a number
t and integers xv..., xN, such that
\tan-bn-xn\ < l/q (n = 1, 2,..., N). (8.3.1)
If all the numbers bn are 0, the result is included in Dirichlet's
theorem. In the general case, we have to suppose the an linearly
independent; for example, if the an are all zero, and the bn are not all
integers, there is in general no t satisfying (8.3.1). Also the theorem
assigns no upper bound for the number t such as the qN of Dirichlet's
theorem. This makes a considerable difference to the results which
can be deduced from the two theorems.
Many proofs of Kronecker's theorem are known.f The following is
due to Bohr A5).
We require the following lemma
Lemma. // <f>(x) is positive and continuous for a ^ x^. b, then
( f U/n
lim U(x)}ndx\ = max ^(x).
a
A similar result holds for an integral in any number of dimensions.
I Bohr A5), A6), Bohr and Jessen C), Estermann C), Lettenmeyer A).

186 Q-THEOREMS Chap. VIII
Let M = max <f>(x). Then
{ J {cf>(x)}n dxYn < {(b—a)MnY'n = {b-aY'nM.
Also, given e, there is an interval, (a, /J) say, throughout which
<f>(x) > M—€.
Hence
6
a
and the result is clear. A similar proof holds in the general case.
Proof of Kronecker's theorem. It is sujBficient to prove that we can find
a number t such that each of the numbers
differs from 1 by less than a given <=; or, if
F(t) =1+2 e27ri'<a»'-6'.>,
n = l
that the upper bound of | F(t) \ for real values of t is 2V-f 1. Let us denote
this upper bound by L. Clearly L ^ N-\-1.
Let GD>v <f>2,..., 4>N) = 1+ f ^^
n =
where the numbers <f>v <f>2,~-, <f>N are independent real variables, each
lying in the interval @,1). Then the upper bound of \G\ is 2V+1, this
being the value of \G\ when fa = <f>2 = ... = <f>N = 0.
We consider the polynomial expansions of {F(t)}k and {G(<f>1,...,<f>N)}k,
where k is an arbitrary positive integer; and we observe that each of
these expansions contains the same number of terms. For, the numbers
av a2,..., aN being linearly independent, no two terms in the expansion
of {F(t)}k fall together. Also the moduli of corresponding terms are
equal. Thus if
..., <l>N)}k =1+2 C3
{F(t)}k =
= 1+
say. Now the mean values
± f \F(t)\"dt
± J
-T

8.3 Q-THEOREMS 187
11 1
and Ok = J J ... J
0 0 0
are equal, each being equal to
This is easily seen in each case on expressing the squared modulus as
a product of conjugates and integrating term by term.
Since JV+1 is the upper bound of \G\, the lemma gives
A'—>-oo
Hence also lim F%2k = JV-f 1.
But plainly F%2k < L
for all values of k. Hence L > 2V+1, and so in fact L = JV+1. This
proves the theorem.
8.4. Theorem 8.4. // a > 1, then
\M\ < W (8-4.1)
for all values of t, while
\Z(8)\ >(l-e)C(a) (8.4.2)
/or some indefinitely large values of t.
We have
n.=
2 n~° = «a),
so that the whole difficulty lies in the second part. To prove this we
use Dirichlet's theorem. For all values of N
?(«) = 2 n-ae~ulogn + f
and hence (the modulus of the first sum being not less than its real part)
- f n~°. (8.4.3)
n = iV+l
By Dirichlet's theorem there is a number t (t0 < t < tQqN) and integers
xv..., xN, such that, for given N and q (q ^ 4),
2tt #t " q
Hence cos(^logn) ^ cosBttjq) for these values of n, and so

188
Hence
Now
and
by (8.4.3)
CO
y
Q-THEOREMS
> oos(&r/,)««)-:
CO
CO /•
^L ' * I ••
n = l J
1
CO
iV + 1
1
Chap. VIII
a—1
Hence |?(s)| > {cos^tt/?) —22V-ff}?(a), (8.4.4)
and the result follows if q and N are large enough.
Theorem 8.4 (A). The function ?(s) is unbounded in the open region
<r >l,t>8 >0.
This follows at once from the previous theorem, since the upper bound
?(cr) of ?(s) itself tends to infinity as a -> 1.
Theorem 8.4 (B). The function ?,(l-\-it) is unbounded as t -> oo.
This follows from the previous theorem and the theorem of Phragmen
and Lindelof. Since ?B-{-it) is bounded, if ?(l-f-^) were also bounded
?(s) would be bounded throughout the half-strip l^cr^2, t > 8; and
this is false, by the previous theorem.
8.5. Dirichlet's theorem also gives the following more precise result.f
Theorem 8.5. However large tx may be, there are values of s in the
region a > 1, t > tv for which
\t(8)\ >A\og\ogt. (8.5.1)
Also ?{l + it) = Q(loglogO- (8.5.2)
Take t0 = 1 and q = 6 in the proof of Theorem 8.4. Then (8.4.4) gives
\C(s)\ >(l-2W-°)l(o-l) (8.5.3)
for a value of t between 1 and 6^. We choose N to be the integer next
above S1^-^. Then
for a value of t such that N > ^41og?. The required inequality (8.5.1)
then follows from (8.5.4). It remains only to observe that the value of t
in question must be greater than any assigned tv if a— 1 is sufficiently
small; otherwise it would follow from (8.5.3) that ?(s) was unbounded
I Bohr and Landau A).

8.5 Q-THEOREMS 189
in the region a > 1, 1 <?<?1; and we know that ?(s) is bounded in
any such region.
The second part of the theorem now follows from the first by the
Phragmen-Lindelof method. Consider the function
ns) = J&L
loglog 5'
the branch of loglog s which is real for 5 > 1, and is restricted to \s\ > 1,
a > 0, t > 0 being taken. Then f(s) is regular for 1 < a < 2, t > 8.
Also [loglog s[ /^ loglog ? as t -> 00, uniformly with respect to a in the
strip. Hence f{2-\-it) -> 0 as t -> 00, and so, if f(l-\-it) -> 0, f(s) -> 0
uniformly in the strip.f This contradicts (8.5.1), and so (8.5.2) follows.
It is plain that arguments similar to the above may be applied to all
Dirichlet series, with coefficients of fixed sign, which are not absolutely
convergent on their line of convergence. For example, the series for
log ?(s) and its differential coefficients are of this type. The result for
log ?(s) is, however, a corollary of that for ?(s), which gives at once
[log ?(«) I > logloglog t—A
for some indefinitely large values of t in a > 1. For the nth. differential
coefficient of log ?(s) the result is that
d\n
>A
for some indefinitely large values of t in a > 1.
8.6. We now turn to the corresponding problems! for l/?(s). We
cannot apply the argument depending on Dirichlet's theorem to this
function, since the coefficients in the series
U) ^ n°
are not all of the same sign; nor can we argue similarly with Kronecker's
theorem, since the numbers (log7i)/2-n- are not linearly independent.
Actually we consider log ?(s), which depends on the series ^p~s, to
which Kronecker's theorem can be applied.
Theorem 8.6. The function l/?(s) is unbounded in the open region
a > 1, t > 8 > 0.
We have for a ^ 1
p m —
mpr
p m = 2 p
P
f See e.g. my Theory of Functions, § 5.63, with the angle transformed into a strip.
X Bohr and Landau G).

190 Q-THEOREMS Chap. VIII
Now
\ oc jV oo
1 \ 'p cos(tlogpn) •sf-* cos(?logpn) v* 1
Also ^e numbers \ogpn are linearly independent. For it follows from the
theorem that an integer can be expressed as a product of prime factors
in one way only, that there can be no relation of the form
Pi'P^-P^ = 1,
where the A's are integers, and therefore no relation of the form
\l°gPi+-~+\N*ogpN = 0.
Hence also the numbers (logpn)/27T are linearly independent. It follows
therefore from Kronecker's theorem that we can find a number t and
integers xv..., xN such that
2tt
or \t\ogpn—7r—27rxn\ < \ir {n = 1, 2,..., N).
Hence for these values of n
TT 2ttX,,) ^ COS^TT = h,
and hence R[ yi-U-if-4- ^ -1
p
Since ^ Pn1 is divergent, we can, if H is any assigned positive number,
choose a so near to 1 that ]? p^CT > H. Having fixed a, we can choose N
so large that N
n=l
Then R( 2H < -i^+iJST = -\H.
Since ^ may be as large as we please, it follows that R(]T p~s), and so
log|?(s) |, takes arbitrarily large negative values. This proves the
theorem.
Theorem 8.6 (A). The function 1/?A -\-it) is unbounded as t -> oo.
This follows from the previous theorem in the same way as Theorem
8.4 (B) from Theorem 8.4 (A).
We cannot, however, proceed to deduce an analogue of Theorem 8.5
for l/?(s). In proving Theorem 8.5, each of the numbers cos(^logn) has
to be made as near as possible to 1, and this can be done by Dirichlet's
theorem. In Theorem 8.6, each of the numbers cos(?logpn) has to be
made as near as possible to — 1, and this requires Kronecker's theorem.

8.6 fi-THEOREMS 191
Now Theorem 8.5 depends on the fact that we can assign an upper limit
to the number t which satisfies the conditions of Dirichlet's theorem.
Since there is no such upper limit in Kronecker's theorem, the corre-
corresponding argument for l/?(s) fails. We shall see later that the analogue
of Theorem 8.5 is in fact true, but it requires a much more elaborate
proof.
8.7. Before proceeding to these deeper theorems, we shall give another
method of proving some of the above results.f This method deals
directly with integrals of high powers of the functions in question, and
so might be described as a short cut which avoids explicit use of
Diophantine approximation. T
We write M{\f(s)\2} = lim -L f |/(a+ti)|2 dt,
T->co 11 J
-T
and prove the following lemma.
Lemma. Let g(s) =>-*?,
/-, ms
m = 1 n = 1
6e absolutely convergent for a given value of a, and let every m with bm ^ 0
6e prime to every n with cn =? 0. Then for such a
M{\g(s)h(s)\>} =
By Theorem 7.1
00 1^ |2
2 1SF-
m = 1 n = 1
= 2 ^s-
Now g(s)h(s) = ^ %
r =
where each term drr~s is the product of two terms bmm~s and cnn~s.
Hence
M{\g(s)h(s)\*} =
r =
We can now prove the analogue for l/?(s) of Theorem 8.4.
Theorem 8.7. // a > 1, then
Tlrr\
(8.7.1)
for all values of t, while
1
1
?(s)
>
for some indefinitely large values of t.
| Bohr and Landau G).

192
Q-THEOREMS
Chap. VIII
We have, for a > 1,
1
Us)
ns
n=l
Since
we have also
n=
and the first part follows.
To prove the second part, write
By repeated application of the lemma it follows that
1 \\2k\
n-
Now, for every p,
m\
1-1
\ _iogP r
)-^ J
1 —
2k
dt,
since the integrand is periodic with period 27r/logp; and
since the Dirichlet series for {r]1\(s)}k begins with 1-{-.... Hence
v 2TTl\ogp
oi2*/ ^ 11 2tt J
dt.
Now lim I
/
0
1-1
pS
2k
112k
Hence
c?^} = max
= 1+-!-.
n
Since the left-hand side is independent of N, we can make N -> oo on
the right, and obtain
. tt°)

8.7 Q-THEOREMS
Hence to any e corresponds a k such that
193
and (8.7.2) now follows.
Since ?(cr)/?Bcr) -> co as a -> 1, this also gives an alternative proof of
Theorem 8.6
It is easy to see that a similar method can be used to prove Theorem
8.4 (A). It is also possible to prove Theorems 8.4 (B) and 8.6 (A) directly
by this method without using the Phragmen-Lindelof theorem. This,
however, requires an extension of the general mean-value theorem for
Dirichlet series.
8.8. Theorem 8.8.f However large t0 may be, there are values of s in
the region a > 1, t > tQfor which
1
i
> A loglog t.
Also
As in the case of Theorem 8.5, it is enough to prove the first part.
We first prove some lemmas. The object of these lemmas is to supply,
for the particular case in hand, what Kronecker's theorem lacks in the
general case, viz. an upper bound for the number t which satisfies the
conditions (8.3.1).
Lemma a. // m and n are different positive integers,
For if m < n
, m
log»
, n . , n
l0gm > l0g^i
max(m, n)
1.11
n
1
n
Lemma j8. Ifpx,..., pN are the first N primes, and nlf..., fxN are integers,
not all 0 (not necessarily positive), then
N
= max|/xn|
N
n=l
For XT Pnn = u/v> wnere
n=l
u = IT p?n> v = TT
<0
f Bohr and Landau G).

194 Q-THEOREMS Chap. VIII
and u and v, being mutually prime, are different. Also
max(w, v) < IT P? < ri^,
n=-l
and the result follows from Lemma a.
Lemma y. The number of solutions in positive or zero integers of the
equation . t
^ + ^+
does not exceed (k-\-\)N.
For N = 1 the number of solutions is k-f-1, so that the theorem holds.
Suppose that it holds for any given N. Then for given vN+1 the number
of solutions of , ,
does not exceed (k—vN+1-\-l)N < (k-\-l)N; and vN+1 can take k-\-l
values. Hence the total number of solutions is ^ (k-\-l)N+1, whence
the result.
Lemma 3. For N > A, there exits a t satisfying 0 ^ t ^ exp(iV6) for
which ,
cos(^logpn) < —1 + Tr (n ^ N).
Let N > 1, k > 1. Then
where c(v0,..., vN) = --—
The number of distinct terms in the expansion is at most (?+1)^ <
by Lemma y. Hence
so that 2 c2 > k~2y( J cf = k~2N(N+lJk.
Let
so that
{F(t)}k =
v,v' ^ n
where Sj is taken over values of (v, v') for which vj = v\, v2 = v2,..., and

8.8 fi-THEOREMS
S2 over the remainder. Now
195
Hence
i f e*
i f | F(t) \
2k dt ^ Sx c2 ~
(a ^ 0).
2cc'
By Lemma jS, since the numbers vn—v'n are not all 0,
^g It Vn n-V;'
1
Hence
M
1 \e j
In this we take k = N*, T = eN\ and obtain, for N > A,
Hence
i J \F(t)\**dt\
Hence there is a ? in @, e-^") such that
__.
\F(t)\>N+l-~.
Suppose, however, that cos(<log^>n) ^ —l-\-ljN for some value of n.
Then
= iV—1 + V2A—
a contradiction. Hence the result.

196 fi-THEOREMS Chap. VIII
We can now prove Theorem 8.8. As in § 8.6, for a > 1
Let now N be large, ( = t(N) the number of Lemma 8, S = I/log N, and
a = 1+8. Then
y cos(dogpn) / 1\^ 1 ^ 1
1 A
->- = AlogN > ^4 loglog*.
The number ? = t(N) evidently tends to infinity with N, since l/?(s) is
bounded in \t\ < A, a > 1, and the proof is completed.
8.9. In Theorems 8.5 and 8.8 we have proved that each of the
inequalities
is satisfied for some arbitrarily large values of t, if A is a suitable
constant. We now consider the question how large the constant can be
in the two cases.
Since neither | ?A -\-it) |/loglog t nor | ?A -{-it) |^/loglog t is known to be
bounded, the question of the constants might not seem to be of much
interest. But we shall see later that on the Riemann hypothesis they
are both bounded; in fact if
loglog t t->co loglog t
then, on the Riemann hypothesis,
(8.9.1)
1 2
A < 2er, /x < —&> (8.9.2)
7T2
where y is Euler's constant.

8.9 fi-THEOREMS 197
There is therefore a certain interest in proving the following results.f
Theorem 8.9 (A). fi^i lj>A+**)l ^ &.
lglgtf
Theorem 8.9 (B).
t-+oo lOglog t 7T2
Thus on the Riemann hypothesis it is only a factor 2 which remains in
doubt in each case.
We first prove some identities and inequalities. As in § 7.19, if
(8-9-3)
and f(x)~ V f(*+m1)!lrcm /« q
and jk{x) - ^|___rj a; , (8.9.
then Fk(s) = UfkiP-)- (8-9.5)
p
Now for real x
fc( ' " 2^ J
r-s-CTl/ilffIC
^ r&—
) t
l
7T
I f ^ I f # /o q 6)
ttJ |1—arle^l2*"- tt J (l-2Va;cos<ii+a;)A:' V " ' '
o o
Using the familiar formula
for the Legendre polynomial of degree n, we see that
Naturally this identity holds also for complex a:; it gives
(8.9.9)
A similar set of formulae holds for l/?(s). We have
| Littlewood E), F), Titchmarsh D), A4).

198 fi-THEOREMS Chap. VIII
Hence _L = V hM (8 9 10)
n = l
where the coefficients bk(n) are determined in an obvious way from the
above product. They are integers, but are not all positive.
The form of these coefficients shows that
Again, let Gk(s) = V ^. (8.9.12)
^—i 71s
tc=1
As in the case of Fk(s),
say. Now, for real x,
k
(8.9.13)
7T IT
= - ( \l+zhi<f>\2kd(f> = - ( (l+2x$cos<f>+x)kd<f>.
TT J TT J
0 0
Comparing this with the formula
IT
Pn(z) = ~ f {z + J(z>-l)C0S<f>}"d<f>
¦n J
o
we see thatf gk(x) = A— x)kPkl^—\. (8.9.14)
Hence
We have also the identity
*). (8-9-15)
t This formula is, essentially, Murphy's well-known formula
Pk(cos8) = cos2*?0.F(-fc, -k; 1; -tan2^)
with x = — tan2?0; cf. Hobson, Spherical and Ellipsoidal Harmonics, pp. 22, 31.

8.9 fi-THEOREMS
Again for 0 < x < \
irlk
d<f>
-L/(-____
it Ik
0
TTlk
0
if k is large enough. Hence also
for k large enough; and
0
for all values of x and k.
8.10. Proof of Theorem 8.9 (A). Let a > 1. Then
J (i-J^-moi** = J (x-^J) J J
K
199
V-^ >t1 + V-) (8917)
- f A + Va;J* ^ = A+VzJ* (8.9.18)
^ = TFkBo). (8.10.1)
Since (from its original definition) fk(p~2a) ^ 1 for all values of p,
-*°) > FT (at1 -pD (8-102)
u
for any positive x and & large enough. Here the number of factors is
7t(x) < ^4.#/log x. Hence if x > V&
(^l2?^) > e-. (8.10.3)
logo;
0

200 fi-THEOREMS Chap. VIII
Also
( J) (
**' V ' ' **' (8.10.4)
Hence FkBa) > e~^-Akia-i)ioex TT A - -1 ,
and
i f/i_iiP
T J \ T)
|l/2fc
as x -> oo, by C.15.2).
Let x = 8k, where k~? < 8 < 1, and a = l + 17/log k, where 0 < t] < 1.
Then the right-hand side is greater than
Also, if mffT = max |?(a+*OI> the left-hand side does not exceed
T * l/2fc
Hence
Let T = iy-4*, so that
loglogT = logA;+logDlog-J.
Then
™<o,t
21?|loglogT-logf41og-jJ.
Giving 8 and 77 arbitrarily small values, and then making k -> 00, i.e.
T -> 00, we obtain
loglog T
where, of course, a is a function of T.

8.10
fi-THEOREMS
201
The result now follows by the Phragmen-Lindelof method. Let
/(*) =
\og\og(s+hi)
I ?(!+»*) I
where h > 4, and let A = lim . .
loglog t
We may suppose A finite, or there is nothing to prove. On a = 1, t ^ 0,
we have
Also, on a = 2, |/E)| = o(l) < A+e (t > tj.
We can choose h so that |/(s)| < A+e also on the remainder of the
boundary of the strip bounded by a = 1, a = 2, and t = 1. Then, by
the Phragmen-Lindelof theorem, |/(s)| < A+e in the interior, and so
=
loglog *
Hence A ^ e^, the required result.
A.
8.11. Proof of Theorem 8.9 (B). The above method depends on the
fact that dk(n) is positive. Since bk(n) is not always positive, a different
method is required in this case.
Let a > 1, and let N be any positive number. Then
f f
1 f
T]
bk(n)
1
T
1 f
TJ
bk(m) ^ bk(n)
Z-,
N
n
2a
I V
i
T
\ I / \ /* / «— \ ¦)*/
I I — I
iana I \m
v \ /
Now
m
n
log
2a
7717^71
w+i ^_i >_L
" 2n ^ 22V'
- dt
2
| log n/m |
n
so that the last sum does not exceed
42V
\bk(m)bk(n)\
00
T Z Z
°n°
m°n
4N
T
IM^)P
= 1
n
T
Since ?(a) ^ l/(a—1) as a-> 1, and ^B) > 1, we have, if a is sufficiently
near to 1, *, % ,
r-1

202 fi-THEOREMS
Hence the above last sum is less than
4:N
Chap. VIII
Also
T(a-lJk'
n8
n>N
n
n>N
for a sufficiently near to 1. Since for a > 2
we have similarly
= 2
n>N
n>N
2 Vk
These two differences are therefore both bounded if
2 \2kKa-l)
N =
[a—lj
With this value of N we have
t t
-f
QfjfcBor) —
-lJk
T(a-l)
by (8.9.17). Now
as in (8.10.4). Hence, as in (8.10.3) and C.15.3),
nf-i+-
1 1 3H va
where b =

8.11 fi-THEOREMS
Choosing x and a as in the last proof,
N
and we obtain
T
203
l2\oglc
T)
4
Finally, let
Then
loglogT =
T =
)
V
V
for k > ^x = ^(e, 77). Hence
I J
0
Let
M T = max
Since the first term on the right of the above inequality tends to infinity
with k (for fixed 8, 77, and e) it is clear that MkT tends to infinity. Hence
?*(*)
<
,T
if k is large enough, and we deduce that
_ log
for A; large enough. Hence
Giving S, e, and 77 arbitrarily small values, and then varying T,
we obtain v
loglog T
The theorem now follows as in the previous case.

204
fi-THEOREMS
Chap. VIII
8.12. The above theorems are mainly concerned with the neighbour-
neighbourhood of the line a = 1. We now penetrate further into the critical strip,
and prove|
Theorem 8.12. Let a be a fixed number in the range | ^ a < 1. Then
the inequality ,y, . .... _ n „.»
y * \C(o+it)\ > exp(loga<)
is satisfied for some indefinitely large values of t, provided that
a < 1 —a.
Throughout the proof k is supposed large enough, and 8 small
enough, for any purpose that may be required. We take \ < a < 1,
and the constants Cv C2,..., and those implied by the symbol 0, are
independent of k and 8, but may depend on a, and on e when it occurs.
The case a = | is deduced finally from the case a > \.
We first prove some lemmas.
Lemma a. Let
in the neighbourhood of s = 1.
|a{*>| < eci* A < m < ifc).
The a^ are the same as those of § 7.13. We have
where
= I cn(s- l)n,
n = 0
n|
I
n ¦-= 0
(C2 > 1, C8 > 1).
Hence e^5 is less than the coefficient of (s— l)n in
Hence
71 = 0
k-m-
71=0
-l)\n\
Lemma p.
00
- ( \T{a
— oo
J
00
—inxe~iS)
| Titchmarsh D).

8.12 fi-THEOREMS
By G.13.3) the left-hand side is greater than
205
00
2 \
-^ dx
ou
J
-1 dx-
00
-2 ("
Since |log(irre~iS)| < log re
\Rk(ixe~*)\ < ^{la
x
and
7r2fc-2a;2a-3
i i
rBJb-l) 7T2fc-2
2A —
—2a'
The result now clearly follows.
Lemma y.
2, aA.(n)exp(—inxe~ld)
J n=--l
x2a~x dx
2
^ 2 d
00 00
The left-hand side is equal to
7
y dk(m)dk(n) exip^mxe18—inxe-^x2**-1 dx
= 2 + 2=2,
00
00
Now
and for 2n sin 8
inS^a-i^ _ Bnsin8)ff f
i
f
2n siu S
00 00
f p-y1l2a-\ rfu ¦> f p-yu2a-l ,]u — C ^> C f>-2nsinS
2n sin o 1
while for 2n sin 8 > 1
00 00
2nsinS
f e-vij2"-1 dy > ( e~y dy =
,-2n sin S

206
Hence
y
Also, using
IS«I < C
1 ^2 1 ^- ^8
— c
— °8
— c
— °8
00
2
n = l
G.14.4),
00
2
m=2
00
v
z
r=l
00
2
r«=l
m —1
2d
00
v
z
m = r+
00
lk(m)d
akym
l
e—^rsinS ^
r
fi-THEOREMS
inxB\nhx*o-i fa > i
p —m sin S
m—n
g-m sin ^
1/7 (nm w\
)U/k\7n —' /
r
0
? dA;(m)e-imsinSd
r+l
Chap. VIII
% VrfINg-2nsinS
n=l
f lyn r\p—\\™— r)sin S
']c\ — /
V rfl(m)e-"»8inS V
= r + l m = r
r
This proves the lemma.
Lemma 8. For a > 1
It is clear from (8.9.6) that
fk(x) < A
Also it is easily verified that
((k+m-l)\y (k2+m-l)
Hence, for 0 < x < 1,
m=0
Hence
2 iogA(p-ff)+ 2
a<A3 p°>k2
2
2 p-tA+olk* 2 p-
o^k* J \ p°>k*
1-a} = 0(k2la).

8.12 fi-THEOREMS 207
On the other hand, (8.10.2) gives
) > 2k J log(l-p-i»)-i- 2 Iog2&
<
^VhC1
p<x log re
>C12lc~-C1*\og2k.
log x log x
Taking x = (^L -±-Y°
the other result follows.
Proof of Theorem 8.12 for \ < a < 1. It follows from Lemmas 0 and
y and Stirling's theorem that
I
00
Now, if 0 < e < 2cr— 1,
00 jo / \ 00
2
(
and
C17
n=l
Let S = exp-^i-°
Then
J
00

208 fi-THEOREMS Chap. VIII
Suppose now that
|?(or+**)l <exp(log«*) (t^t0)
where 0 < a < 1. Then
00 00
ify-i dt
f f
J ^ 19+J
0
If t > &2/S2, k > fc0, then
Hence
8 ^ ^ 2 log"**
g
00
/
J e2fcl0B«<-2^2a-l ^ < eSA.-loB-tftVS*) J e-2^2a-l ^_|_ J e-
l l fc«/S«
Hence -^- = Olklog*^) = O(Jfci+B«VBa-€))#
\logfc/ \ 8/
8/
Hence - < 1-] ,
a 2a—e
and since e may be as small as we please
1 a
a a
The case a = \. Suppose that
?(*+#) = O{exp(log^)} @ < )8 < i).
Then the function /(s) = ^E)exp(—log^s)
is bounded on the lines a = \, a = 2, t > t0, and it is 0(?) uniformly
in this strip. Hence by the Phragmen-Lindelof theorem f(s) is bounded
in the strip, i.e.
?(+it)
for I < a < 2. Since this is not true for \ < a < 1 —y3, it follows that
NOTES FOR CHAPTER 8
8.13. Levinson [1 ] has sharpened Theorems 8.9(A) and 8.9(B) to show
that the inequalities
ICU + 101 > cMoglog t + O(l)

8.13 fi-THEOREMS 209
and
1 6ey
(loglog t - logloglog 0 + O(l)
each hold for arbitrarily large t. Theorem 8.12 has also been improved,
by Montgomery [3]. He showed that for any a in the range \ < a < 1,
and for any real S, there are arbitrarily large t such that
Here log C(s) is, as usual, defined by continuous variation along lines
parallel to the real axis, using the Dirichlet series A.1.9) for a > 1. It
follows in particular that
and the same for ^(o + ii)-1. For o = \ the best result is due to
Balasubramanian and Ramachandra [2], who showed that
max ICQ+iOl^expff ¦
T<t<T+H V (log log H)*/
if (log T)d ^H^ Tand T ^ T(S), where ^ is any positive constant. Their
method is akin to that of § 8.12, in that it depends on a lower bound for a
mean value of \(,(\ + ii)\2k, uniform in k. By contrast the method of
Montgomery [3] uses the formula
(logO2
4 I /sini-vX^
e-^logtiG+it + iy)^ ^-j {1 + cos(9+ ylogx)}dy
-OogO2
= Z ^-n-°-i'(%-\\og-\) + O{x(\ogt)-2}. (8.13.1)
This is valid for any real x and S, providing that ?(s) ^ 0 for R(s) ^ a and
| I(s) —1| < 2(log t J. After choosing x suitably one may use the extended
version of Dirichlet's theorem given in §8.2 to show that the real part of
the sum on the right of (8.13.1) is large at points tx < ... <tN^ T, spaced
at least 4(log TJ apart. One can arr.ange that N exceeds N(o, T),
whence at least one such tn will satisfy the condition that ?(s) =? 0 in the
corresponding rectangle.

IX
THE GENERAL DISTRIBUTION OF THE ZEROS
9.1. In § 2.12 we deduced from the general theory of integral functions
that ?(s) has an infinity of complex zeros. This may be proved directly
as follows.
We have
- + -+ <- + — + ^-+ =I-i-/I_lWfi-iW --
Hence for a > 2
and
Also R{CW}= l + C-^«-) + ... > 1-1-... >i. (9.1.3)
Hence for o- > 2 we may write
log?(s) = log |CE) I+i arg ?(s),
where arg ^E) is that value of arctan{I?(s)/R?(,s)} which lies between
—\n and \tt. It is clear that
I log ?(«)|< 4 (or>2). (9.1.4)
For c < 2, t ^ 0, we define log ?(s) as the analytic continuation of
the above function along the straight line {a-{-it, 2-{-it), provided that
?(s) ^Oon this segment of line.
Now consider a system of four concentric circles Cx, C2, C3, C4, with
centre 3+*^ and radii 1, 4, 5, and 6 respectively. Suppose that ?(s) 7^ 0
in or on C4. Then log ?(s), defined as above, is regular in C4. Let Mlf
Mz, Mz be its maximum modulus on Clt Cz, and C3 respectively.
Since ?(s) = 0{tA), R{log ?(s)} < ^ log T in C4, and the Borel-
Caratheodory theorem gives
^± <A\ogT.
Also Hx < A, by (9.1.4). Hence Hadamard's three-circles theorem,
applied to the circles Clt C2, C3, gives
Mt^MlM% <A\ogPT,
where 1—a = jS = Iog4/log5 < 1.

9.1 GENERAL DISTRIBUTION OF ZEROS 211
Hence ?(—1+iT) = O{exp(log^T)} = 0{T<).
But by (9.1.2), and the functional equation B.1.1) with a = 2,
We have thus obtained a contradiction. Hence every such circle C4
contains at least one zero of ?(s), and so there are an infinity of zeros.
The argument also shows that the gaps between the ordinates of
successive zeros are bounded.
9.2. The function N(T). Let T > 0, and let N{T) denote the
number of zeros of the function ?(s) in the region 0 < o- ^ 1, 0 < t ^ T.
The distribution of the ordinates of the zeros can then be studied by
means of formulae involving N(T).
The most easily proved result is
Theorem 9.2. As T -> oo
N{T+l)-N{T) = O{\ogT). (9.2.1)
For it is easily seen that
N(T+1)—N(T) <n(V5),
where n(r) is the number of zeros of ?(s) in the circle with centre 2-\-iT
and radius r. Now, by Jensen's theorem,
3 2tt
j!!<pdr = l.[ log\?B+iT+3e*°)\ ^-log|«2+*T)|.
o o
Since \?,{s)\ < tA for — 1 ^ a < 5, we have
log|?B+*T+3e<°)| <A\ogT.
3
Hence (^dr <A\og T+A < A logT.
o
3 3 3
Since [V^ldr^ [V^Ldr^ n(V5) f ^ = 4n(V5),
0 V5 V5
the result (9.2.1) follows.
Naturally it also follows that
N{T+h)-N{T) = O{\ogT)
for any fixed value of h. In particular, the multiplicity of a multiple
zero of ?(s) in the region considered is at most O(log T).

212 GENERAL DISTRIBUTION OF ZEROS Chap. IX !
9.3. The closer study of N(T) depends on the following theorem.f
If T is not the ordinate of a zero, let S(T) denote the value of
obtained by continuous variation along the straight lines joining 2,
2-\-iT, \-\-iT, starting with the value 0. If T is the ordinate of a zero,
let 8{T) = 8{T+0). Let
L(T) = ±TlogT-l±^T + Z. (9.3.1)
Theorem 9.3. As T -> oo
N{T) = L{T) + S(T) + OA/T). (9.3.2)
The number of zeros of the function S(z) (see § 2.1) in the rectangle
with vertices at z = ±T±\i is 2N(T), so that
taken round the rectangle. Since H(z) is even and real for real z, this
is equal to
T+ii |i o+iT i-HT.
K7
TT
where A denotes the variation from 2 to 2-\-iT, and thence to \-\-iT,
along straight lines. Recalling that
we obtain
Now Aarga(a-l) = arg(-|-T2) = rr,
AargTr-^ — Aarge-2i>log7r = —
and by D.12.1)
Adding these results, we obtain the theorem, provided that T is not the
ordinate of a zero. If T is the ordinate of a zero, the result follows from
f Backlund B), C).

9.3 GENERAL DISTRIBUTION OF ZEROS 213
the definitions and what has already been proved, the term 0A/T) being
continuous.
The problem of the behaviour of N(T) is thus reduced to that of S(T).
9.4. We shall now prove the following lemma.
Lemma. Let 0 ^ a. < /? < 2. Let f(s) be an analytic function, real for
real s, regular for a ^ a. except at s = 1; let
\RfB+it)\ >ra > 0
and l/(<*'+*OI < Mai {a ^ a, 1 < t' < t).
Then if T is not the ordinate of a zero of f(s)
for a > ?.
Since arg/B) = 0, and
where R/(s) does not vanish on a = 2, we have
|arg/B+*T)| <\rr.
Now if R/(s) vanishes q times between 2+iT and fi-\-iT, this interval
is divided into q-\-1 parts, throughout each of which R{/(s)} > 0 or
R{/(s)} < 0. Hence in each part the variation of arg/(s) does not
exceed -n. Hence [arg/E)| ^ {q+i)n {<J > ^
Now q is the number of zeros of the function
for I(z) = 0, /? ^ R(z) ^ 2; hence q ^ n{2—/?), where n(r) denotes the
number of zeros of g(z) for [z—2| ^ r. Also
2—a 2—a
J
0 2-j3
and by Jensen's theorem
2 —a 2tt
0
This proves the lemma.

214 GENERAL DISTRIBUTION OF ZEROS Chap. IX
We deduce
Theorem 9.4. As T -> oo
S(T) = O(\ogT), (9.4.2)
i.e. N(T) = J-yiogy-1+1o0g27ry+0(logy). (9.4.3)
We apply the lemma with f(s) = ?(s), a = 0, j8 = ?, and (9.4.2)
follows, since ?(s) = O(^). Then (9.4.3) follows from (9.3.2).
Theorem 9.4 has a number of interesting consequences. It gives
another proof of Theorem 9.2, since @ <0 < 1)
L(T+l)-L{T) = L'{T+0) = O(\ogT).
We can also prove the following result.
// the zeros fi-\-iy of ?(s) with y > 0 are arranged in a sequence
Pn = /?n+*Vn so that yn+1 ^ yn, then as n -> oo
We have N{T) ~ -- Tlog T.
Hence 27rN(yn± 1) ~ (yn ± 1)\og(yn±1) ~ ynlogyn.
Also N(yn—1) ^ n ^ N(yn-\-l).
Hence 2rrn ~ yn log y„..
Hence logn --^ logyri,
27771
and so yn ^ -, •
log n
Also \pn\~Yn, since j8n = 0A).
We can also deduce the result of § 9.1, that the gaps between the
ordinates of successive zeros are bounded. For if \S(t) \ ^ Clogt (t > 2),
T+H
N(T+H)-N(T) = ±t f
which is ultimately positive if H is a constant greater than
The behaviour of the function S(T) appears to be very complicated.
It must have a discontinuity k where T passes through the ordinate of
a zero of ?(s) of order k (since the term 0{ljT) in the above theorem is
in fact continuous). Between the zeros, N{T) is constant, so that the

9.4 GENERAL DISTRIBUTION OF ZEROS 215
variation of S(T) must just neutralize that of the other terms. In the
formula (9.3.1), the term g is presumably overwhelmed by the variations
of S(T). On the other hand, in the integrated formula
T T T
J N{t) dt = J L{t) dt+ J S(t) dt+0(\og T)
0 0 0
the term in S{T) certainly plays a much smaller part, since, as we shall
presently prove, the integral of S(t) over @, T) is still only 0(log T).
Presumably this is due to frequent variations in the sign of S(t). Actually
we shall show that S(t) changes sign an infinity of times.
9.5. A problem of analytic continuation. The above theorems on
the zeros of ?(s) lead to the solution of a curious subsidiary problem of
analytic continuation.f Consider the function
. (9.5.1)
P
This is an analytic function of 5, regular for a > 1. Now by A.6.1)
*—i n
n = l
As n -> oo, log ?,(ns) ~ 2-713. Hence the right-hand side represents an
analytic function of s, regular for a > 0, except at the singularities of
individual terms. These are branch-points arising from the poles and
zeros of the functions ?(ns); there are an infinity of such points, but they
have no limit-point in the region a > 0. Hence P(s) is regular for
a > 0, except at certain branch-points.
Similarly, the.function
Q(8) = -
is regular for a > 0, except at certain simple poles.
We shall now prove that the line a = 0 is a natural boundary of the
functions P(s) and Q{s).
We shall in fact prove that every point of a = 0 is a limit-point of
poles of Q{s). By symmetry, it is sufficient to consider the upper half-
line. Thus it is sufficient to prove that for every u > 0, 8 > 0, the
square 0 < a < S, u < t < u+Z (9.5.4)
contains at least one pole of Q(s).
f Landau and Waliisz A).

216 GENERAL DISTRIBUTION OF ZEROS Chap. IX
As p -> oo through primes,
N{p{u+8)} ~ —- {u+h)plogp, N{pu)~ — up\ogp,
ATT LTT
by Theorem 9.4. Hence for all p ^ po(8, u)
N{p{u+h)}-N{pu) > 0. (9.5.5)
Also, by Theorem 9.2, the multiplicity v(p) of each zero p = j8-f iy with
ordinate y > 2 is less than A logy, where A is an absolute constant.
Now choose p = p(8, w) satisfying the conditions
1 2
P>*> P^-> P>Po@,u), p > A\og{p{u+S)}.
There is then, by (9.5.5), a zero p of ?(s) in the rectangle
? < a < 1, pu <t ^ p(u+h). (9.5.6)
Since y > pw ^ 2, its multiplicity v(p) satisfies
v(p) < A logy < A log{p(^+8)} < p,
and so is not divisible by p.
The point p/p belongs to the square (9.5.4). We shall show that this
point is a pole of Q{s). Let m run through the positive integers (finite
in number) for which ?,(mp/p) = 0. Then we have to prove that
m \p I
The term of this sum corresponding to m = p is —v(p)/p. No other m
occurring in the sum is divisible by p, since for m ^ 2,p
(mp\ =='^>
R(mp
Hence ^J
m \ p / o p
where a and 6 are integers, and p is not a factor of b. Since p is also
not a factor of v(p), ap ^ bv(p), and (9.5.7) follows.
There are various other functions with similar properties. For
examples let ^ = -
n = l
where k and Z are positive integers, k > 2. By A:2.2) and A.2.10),
/IA.(s) is a meromorphic function of 5 if I = 1, or if I = 2 and & = 2.
i^or oZZ other values of I and k, fitk(s) has a = 0 as a natural boundary,
and it has no singularities other than poles in the half-plane a > 0.
f Estennann A).

9.6 GENERAL DISTRIBUTION OF ZEROS 217
9.6. An approximate formula for ?'(s)/?(s). The following approxi-
approximate formula for ?'(s)/?(s) in terms of the zeros near to s is often useful.
Theokem 9.6 (A). If p = j8+ty runs through zeros of ?(s),
uniformly for — 1 < a ^ 2.
Take /(s) = ?(s), s0 = 2+iT, r = 12 in Lemma a of § 3.9. Then
M = A log jP, and we obtain
—+ O(logT) (9.6.2)
-/>
for ]s—50| ^ 3, and so in particular for —1 < a ^ 2, ? = T. Replacing
T by ^ in the particular case, we obtain (9.6.2) with error O(\ogt), and
— 1 ^ a ^ 2. Finally any term occurring in (9.6.2) but not in (9.6.1) is
bounded, and the number of such terms does not exceed
by Theorem 9.2. This proves (9.6.1).
Another proof depends on B.12.7), which, by a known property of
the F-function, gives
Jut = 2 (tit+-)+o(log*)•
Replacing s by 2-{-it and subtracting,
since t>\2+it)it,{2+it) = 0A).
Now
by Theorem 9.2. Also
/I 1 \ _ ^ 2—a
\s—p~2+it—p) ~~ ^-i ls—p)B+it—p)
- 2 <feM- 2
<+n<y<<+n+l u/ ; ; < + <
again by Theorem 9.2. Since
y logit+n) y tog« y Iog2n =
Z-y 7i2 Z-y n* ^—1 n~
Z-y 7i y n 1
n=l n<< n>t

218 GENERAL DISTRIBUTION OF ZEROS Chap. IX
it follows that V [— \ ) = Olloet).
y>t+l v r i r/
Similarly ? fe-2+k"J = 0(U*()
and the result follows again.
The corresponding formula for log ?(s) is given by
Theokem 9.6 (B). We have
I 0 (9.6.3)
uniformly for — 1 ^ a ^ 2, where log ?(s) has its usual meaning, and
— 77 < IlOg(S — p) < 7T.
Integrating (9.6.1) from s to 2-{-it, and supposing that t is not equal
to the ordinate of any zero, we obtain
log?(*)-log?B+t«)= 2 {log(*-p)-logB+tt-p)}+0(log«).
\t-y\<l
Now log^B4-*i) is bounded; also IogB-fi2—p) is bounded, and there
are O(logf) such terms. Their sum is therefore 0(log?). The result
therefore follows for such values of t, and then by continuity for all
values of 5 in the strip other than the zeros.
9.7. As an application of Theorem 9.6 (B) we shall prove the following
theorem on the minimum value of ?(s) in certain parts of the critical
strip. We know from Theorem 8.12 that |?(s)| is sometimes large in
the critical strip, but we can prove little about the distribution of the
values of t for which it is large. The following resultf states a much
weaker inequality, but states it for many more values of t.
Theokem 9.7. There is a constant A such that each interval (T, T-{-l)
contains a value of t for which
\Z(s)\>t-A (-l<a<2). (9.7.1)
Further, if H is any number greater than unity, then
\?{8)\ > T~AH (9.7.2)
for — 1 =^ a < 2, T ^ t ^ T-{-l, except possibly for a set of values of t
of measure 1/H.
Taking real parts in (9.6.3),
I g (9.7.3)
\t-y\<l
f Valiron A), Landau (8), A8), Hoheisel C).

9.7 GENERAL DISTRIBUTION OF ZEROS 219
Now
(yl,+l)
f 2 \ag\t-y\dt= 2 f \og\t-y\dt
> I I
2 (~2)>-A\QgT.
^r+2
Hence Y log|?—y| > — ^4 log T
l«-yl<l
for some fin (T,
Hence logics)| > —^logT for some t in {T,T+1) and all a in
— 1 < a < 2; and
except in a set of measure l/H. This proves the theorem.
The exceptional values of t are, of course, those in the neighbourhood
of ordinates of zeros of ?(s).
9.8. Application to a formula of Ramanujan.f Let a and b be
positive numbers such that ab = n, and consider the integral
27rtJ J(l-2«) 2Sj
taken round the rectangle (lib*^7, — \^T). The two forms are equiva-
equivalent on account of the functional equation.
Let T ->¦ oo through values such that | T—y\ > exp(—^41y/logy) for
every ordinate y of a zero of ?(s). Then by (9.7.3)
where ^42 < \tt if ^4! is small enough, and T > T7,,. It now follows from
the asymptotic formula for the F-function that the integrals along the
horizontal sides of the contour tend to zero as T -> oo through the above
values. Hence by the theorem of residues %
. -i+xoo 1 + xoo
2-ni J ?(l-2«) 2iri J Vtt JB«)
— J — 100 1 — 100
f Hardy and Littlewood B), 156-9.
j In forming the series of residues we have supposed for simplicity that the zeros of ?(«)
are all simple.

220 GENERAL DISTRIBUTION OF ZEROS Chap. IX
The first term on the left is equal to
-J + ioo
?-, n 2-m J \a
_e-(a/nJl
n =
Evaluating the other integral in the same way, and multiplying through
by Va, we obtain Ramanujan's result
n =
(9.8.1)
We have, of course, not proved that the series on the right is con-
convergent in the ordinary sense. We have merely proved that it is conver-
convergent if the terms are bracketed in such a way that two terms for which
\y—y'\ < exp(—^1y/logy)+exp(-^1y71ogy')
are included in the same bracket. Of course the zeros are, on the average,
much farther apart than this, and it is quite possible that the series may
converge without any bracketing. But we are unable to prove this,
even on the Riemann hypothesis.
9.9. We next prove a general formula concerning the zeros of an
analytic function in a rectangle.f Suppose that <f>(s) is meromorphic in
and upon the boundary of a rectangle bounded by the lines t = 0,
t = T, a = a, a = j8 (j8 > a), and regular and not zero on a = j8. The
function log(/>(s) is regular in the neighbourhood of a = j8, and here,
starting with any one value of the logarithm, we define F{s) = log <f>(s).
For other points 5 of the rectangle, we define F(s) to be the value
obtained from Iog</>(j8-|-^) by continuous variation along t = constant
from j8+^ to a-{-it, provided that the path does not cross a zero or
pole of <f>{s); if it does, we put
F(s) = lim F{o+it+ie).
Let v{a', T) denote the excess of the number of zeros over the number
of poles in the part of the rectangle for which a > a', including zeros
or poles on t = T, but not those on t = 0.
Then J F{s) ds = -2ni \ v{a, T) da, (9.9.1)
a
the integral on the left being taken round the rectangle in the positive
direction.
f Littlewood D).

9.9 GENERAL DISTRIBUTION OF ZEROS 221
We may suppose t = 0 and t = T to be free from zeros and poles of
(f>(s); it is easily verified that our conventions then ensure the truth of
the theorem in the general case.
We have
B B T
J F(s) ds = J F{a) da- J F(a+iT) da+ J {F(p+it)-F(*+it)} idt.
a a 0
(9.9.2)
The last term is equal to
T B B o+iT
* rfM !i/ 7v\ da= \ da ^ ds,
J J cf>(a+it) J J <f>(s)
0 a a a
and by the theorem of residues
o + iT , B B + iT B + iT,
a a B o+iT '
Substituting this in (9.9.2), we obtain (9.9.1).
We deduce
T
Theokem 9.9. // SX{T) = J 8{t) dt,
o
2
then S^T) = - [ \og\t,{a+iT)\ da+O(l). (9.9.3)
* J
Take (f>{s) = ^(s), a. = \, in the above formula, and take the real part.
We obtain
B T B
J log! ?(*) | da- J arg ?(P+it) dt- j log| t,{a+iT) \ da+
i 0
2
+ J arg?(l+**)<ft= 0, (9.9.4)
o
the term in v(a, T), being purely imaginary, disappearing. Now make
j8 -> oo. We have
as a -> oo, uniformly with respect to f. Hence arg ?(s) = 0B~a), so that
the second integral tends to 0 as j8 -> oo. Also the first integral is a
constant, and
J log|S(<r+*3T)| rfa = J 0B-') ^a = 0A)

222 GENERAL DISTRIBUTION OF ZEROS Chap. IX
Hence the result.
Theorem 9.9 (A). S^T) = O(log T).
By Theorem 9.6 (B)
2 2
l°gl?E)l da = y log|s—p\ da-^-OCiogt).
The terms of the last sum are bounded, since
2 2
> J log{(a-i8J+(y-02} rfcr > 2 J Iog|a-j8|
2
Hence J logics)| ^a = O(logf), (9.9.5)
i
and the result follows from the previous theorem.
It was proved by F. and R. Nevanlinna A), B) that
jMdt = A + 0^.). (9.9.6)
0
This follows from the previous result by integration by parts; for
11 T
Since SX{T) = O(log T), the middle term is O(T-1log7T), and the last
term is x
Hence the result follows. A similar result clearly holds for
T
j^dt @<«<l).
i
It has recently been proved by A. Selberg E) that
S(t) = Q±{(log«)*(loglog«)-*} (9.9.7)
with a similar result for S^t); and that
S^t) = n+{(log0^(loglogf)-4}. (9.9.8)
9.10. Theorem 9.10.f S(t) has an infinity of changes of sign.
Consider the interval (yn,yn+1) in which N(t) = n. Let l(t) be the
f Titchmarsh A7).

9.10 GENERAL DISTRIBUTION OF ZEROS 223
linear function of t such that l(yn) = S(yn), l(yn+1) = S(yn+1 — 0). Then
for yn < t < yn+1
W-S(t) = {±Z
\y«/
using (9.3.2) and the fact that N(t) is constant in the interval. The first
two terms on the right give
-L'(?){t-yn)+L'{V)(t-yn) (yn <V<t,yn<?< yn+1)
between ? and v)
since ytt+1—ytt = 0A). Hence
y«+i y«+i
I S(t)dt= f
Suppose that S(t) > 0 for t > ?0. Then
gives
Hence y»+
J
Hence J S{t) dt^ \{ys-yj,
y»0
contrary to Theorem 9.9 (A). Similarly the hypothesis S(t) < 0 for
t > t0 can be shown to lead to a contradiction.
It has been proved by A. Selberg E) that S(t) changes sign at least
T(logT)le-A"losloeT
times in the interval @, T).
9.11. At the present time no improvement on the result
S(T) = O(\ogT)
is known. But it is possible to prove directly some of the results which
would follow from such an improvement. We shall first provef
Theorem 9.11. The gaps between the ordinates of successive zeros of
t,{s) tend to 0.
t Littlewood C).

224 GENERAL DISTRIBUTION OF ZEROS Chap. IX
This would follow at once from (9.3.2) if it were possible to prove that
The argument given in § 9.1 shows that the gaps are bounded. Here
we have to apply a similar argument to the strip T—8 ^C t ^C T-{-h,
where 8 is arbitrarily small, and it is clear that we cannot use four
concentric circles. But the ideas of the theorems of Borel-Caratheodory
and Hadamard are in no way essentially bound up with sets of concentric
circles, and the difficulty can be surmounted by using suitable elongated
curves instead.
Let ZL be the rectangle with centre 3 -f- i T and a corner at— 3-{-i(T-{-&),
the sides being parallel to the axes. We represent D4 conformally on the
unit circle D4 in the z-plane, so that its centre 3+^ corresponds to
z = 0. By this representation a set of concentric circles \z\ = r inside
D4 will correspond to a set of convex curves inside D4, such that as
r -> 0 the curve shrinks upon the point 3-{-iT, while as r -> 1 it tends
to coincidence with D4. Let D\, D'2, D'3 be circles (independent, of course,
of T) for which the corresponding curves Dlf D2, D3 in the s-plane pass
through the points 2-{-iT, —1+iT, — 2-{-iT respectively.
The proof now proceeds as before. We consider the function
f(z) = log Z{s(z)},
where s — s(z) is the analytic function corresponding to the conformal
representation; and we apply the theorems of Borel-Carathe"odory and
Hadamard in the same way as before.
9.12. We shall now obtain a more precise result of the same kind.f
Theokem 9.12. For every large positive T, ?(s) has a zero
satisfying a
\y-T\ <
logloglogT'
This was first proved by Littlewood by a detailed study of the con-
formal representation used in the previous proof. This involves rather
complicated calculations with elliptic functions. We shall give here two
proofs which avoid these calculations.
In the first, we replace the rectangles by a succession of circles. Let
The a large positive number, and suppose that ?(s) has no zero j8-f-*V
such that T—8 < y < T+8, where 8 < $. Then the function
/(*) = log ?(8),
where the logarithm has its principal value for a > 2, is regular in the
rectangle -2 < a < 3, T-B < t < T+8.
f Littlewood C); proofs given here by Titchmarsh A3), Kramaschke A).

9.12 GENERAL DISTRIBUTION OF ZEROS 225
Let cv, Cv, Cv, Vv be four concentric circles, with centre 2—\vh-\-iT, and
radii \h, \h, §8, and 8 respectively. Consider these sets of circles for
v = 0, 1,..., n, where n = [12/8]+1, so that 2—\nh < — 1, i.e. the
centre of the last circle lies on, or to the left of, a = — 1. Let mv, Mv,
and Mv denote the maxima of \f(s) | on cv, Cv, and Cv respectively.
Let Ax, A2,... denote absolute constants (it is convenient to preserve
their identity throughout the proof). We have R{/(s)} < A^ogT on
all the circles, and |/B+i^T)| < A2. Hence the Borel-Caratheodory
theorem for the circles Co and ro gives
Mo <|±g(^1logT+i42) = HAilogT+AJ,
and in particular
\fB-±B+iT)\ <7(A1\ogT+A2).
Hence, applying the Borel-Caratheodory theorem to Cx and Tx,
Mx < 1{AXlogT+1/B-lS+iT)\} <
So generally Mv < G + ... + 7"+1)^41log
or, say, Mv < 7vA3\ogT. (9.12.1)
Now by Hadamard's three-circles theorem
where a and b are positive constants such that a-\-b = 1; in fact
a = log |/log 3,6 = log 2/log 3. Also, since the circle Cv_x includes the
circle cv, mv < Mv_x. Hence
Mv^Mav_xMbv (v= 1,2,..., 7i).
Thus 3I1^M^Mb1, 3f2 ^ MfM* ^ MtfM?Mb2,
and so on, giving finally
Mn < MfMf'lbMf'2b..Mbn.
Hence, by (9.12.1),
M <
Now a"-16+2a"-26+...+7i6 < n\
an~16+an-26+...+6 = 6A—an)/(l— a) = l—an.
Hence Mn < if f7n2(^3 log TI-"" < Ax 7(log TI-"",
since i/0 is bounded as T -.> oo.
But |^(s)| > tA* for a < — 1, t > t0, so that Mn > ^45log T. Hence
loglogT
logloglog T <n log - + ,4 6 log n,
CL

9.13 GENERAL DISTRIBUTION OF ZEROS 227
A
Hence log M2 < A + Xo^lr-1 \0
Now if rx, r2, and r3 are sufficiently near to 1, i.e. if a is sufficiently small,
logfl + ^!i) ,_ __xi
i | r
-rt l-r2
and r^-r, = 1+r, l+r2 l+r2
l 1-^ l+r3
x l+r3
Hence
Also 1/A—r3) <
Hence log Jtf2 < ^4 + {l-^4(|)^a}floglog T + -log 5+AJ.
Let a = 7r/(clogloglog T). Then
f}{loglog T+clog5logloglog T+A}
if clog| < ? and T is large enough. Hence
M2 < \ogTe-^°^osT)i < e lOg T (T > T0(e)).
In particular log|^( - 1+»!T)|< e log T,
But
We thus obtain a contradiction, and the result follows.
9.14. Another resultf in the same order of ideas is
Theoeem 9.14. For any fixed h, however small,
N(T+h)-N(T) > KlogT
for K = K(h), T > To.
This result is not a consequence of Theorem 9.4 if h is less than a
certain value.
Consider the same angular region as before, with a new a such that
f Not previously published.

228
GENERAL DISTRIBUTION OF ZEROS
Chap. IX
tana < ?, and suppose now that ?(s) has zeros plt p2,..., pn in the
angular region. Let
F(s) =^
— Pl)-(S — pn)
Let C be the circle with centre %-\-iT and radius 3. Then \s—pv
onC. Hence \F(s)\ ^ |?(«)| < T*
on C, and so also inside C.
Let/(s) = log-F(s). Then/(«s) is regular in the angle, and
R/(s) < A log T.
Also
fB+iT) =
n
v=l
»T)- 2 logB+iT-Pv)
v=l
= 0(n).
Let ifcfu ifcf2, and Mz now denote the maxima of \f(s)\ on the three
s-curves. Then *
Also i^ < An, as for/B+iT). Hence
< logif2
<
{A +log n
logf 4^±l°
< ^4 + log*
1-^
//
as before. But
n
say. If 7i > ^(^i/^42) l°g ^ *ne theorem follows at once. Otherwise
and we obtain
n
-^ (*)»/«}!! log 5,
a

9.14 GENERAL DISTRIBUTION OF ZEROS 229
and hence loglog/-^—j <- log- + log—\-A < —,
\ n J a 4 a a
n
This proves the theorem.
9.15. The function N(a, T). We define N(a, T) to be the number
of zeros fi-\-iy of the zeta-function such that /? > a, 0 < t ^ T. For
each T, N(a, T) is a non-increasing function of a, and is 0 for a > 1.
On the Riemann hypothesis, N(a, T) ~ 0 for a > ?. Without any
hypothesis, all that we can say so far is that
N{a, T) < N(T) < ATlogT
for J < a < 1.
The object of the next few sections is to improve upon this inequality
for values of a between \ and 1.
We return to the formula (9.9.1). Let <f>(s) = ?(s), a = a0, /J = 2,
and this time take the imaginary part. We have
v{a,T) = N{a,T) (a < 1), v(a, T) = 0 (or ^ 1).
We obtain, if T is not the ordinate of a zero,
IT T
2tt J tf(or, T) da = J log |?(or0+*OI <«- J log |?B+*0|
CT0
where ^L(a0) is independent of T. We deducef
Theobem 9.15. Ij\ < a0 < 1, anci T ->oo,
2,7 J i\T(aj T) da = J log| ?(or0+*01 *+ O(log iT).
CT0 0
We have
Also, by § 9.4, arg^(a+iT) = O(log T) uniformly for a ^ \, if T is
not the ordinate of a zero. Hence the integral involving arg?(cr+iT)
is O(log T). The result follows if T is not the ordinate of a zero, and
this restriction can then be removed from considerations of continuity.
t Littlewood D).

226 GENERAL DISTRIBUTION OF ZEROS Chap. IX
so that 8 < < —— -,
Ti—1 logloglog T
and the result follows.
9.13. Second Proof. Consider the angular region in the s-plane
with vertex at s = — Z-\-iT, bounded by straight lines making angles
+ ^a@ < a < n) with the real axis.
Let w = (s+3—iTyi*.
Then the angular region is mapped on the half-plane ~R(w) ^ 0. The
point s = 2-\-iT corresponds to
w = 5Trloc.
Let z =
Then the angular region corresponds to the unit circle in the z-plane,
and s = 2-\-iT corresponds to its centre z = 0. If s = a-\-iT corre-
corresponds to z = —r, then
= w =
—-,
1+r
i.e. r = {l —
Suppose that ?(s) has no zeros in the angular region, so that log ?(s) is
regular in it.
Let s = %-\-iT, — \-\-iT, —2-\-iT correspond to z = —rx, — r2, —r3
respectively. Let Mx, M2, Mz be the maxima of |log t,{s) \ on the s-curves
corresponding to \z\ = rv r2, r3. Then Hadamard's three-circles theorem
glV6S logM2 < p^HogM1 + p^\ogM3.
g 2^ log rjr, * ^ log rjr, s 3
It is easily verified that, on the curve corresponding to \z\ = rx,
a ^ f. For if w = i-^-irj, then
a= —
which is a minimum at 17 = 0, for given ?, if 0 < a < \n\ and the
minimum is — 3+?a/71", which, as a function of ?, is a minimum when ?
is a minimum, i.e. when z = —rx. It therefore follows that logi^ < A.
Since R{log ?(s)} < A log T in the angle, it follows from the Borel-
Caratheodory theorem that
M3 < (A log T+A) <
' 3

230 GENERAL DISTRIBUTION OF ZEROS
Theobem 9.15(A).f For any fixed a greater than \,
N(a, T) = 0{T).
For any non-negative continuous f(t)
( i r
dt
Chap. IX
b-i
U
a
a
Thus, for | < a < 1,
T T
j log\ U°+it)\ dt = % j \og\ tta+it)\* dt
0 0
. T
^ 0
by Theorem 7.2. Hence, by Theorem 9.15,
J N(a, T) da = O(T)
for a0 > \. Hence, if ax = !+!(cr0—i)>
Co 1
1 f 2 f
^ N(a, T) da ^ -
= O(T)
CTj
the required result.
From this theorem, and the fact that N(T) ~ AT log T, it follows
that all but an infinitesimal proportion of the zeros of ?(s) lie in the strip
\—8 < a < ?+3, however small 8 may be.
9.16. We shall next prove a number of theorems in which the O(T)
of Theorem 9.15 (A) is replaced by O(Te), where 0 < l.J We do this by
applying the above methods, not to ?(s) itself, but to the function
n<X
ns
The zeros of ?(s) are zeros of ?,(s)Mx(s). If a > 1, Mx(s) -> l/?(s) as
X -> oo, so that ?,{s)Mx(s) -> 1. On the B-iemann hypothesis this is also
true for \ < a ^ 1. Of course we cannot prove this without any
hypothesis; but we can choose X so that the additional factor neutralizes
to a certain extent the peculiarities of ?(s), even for values of a less than 1.
Let fx(s) = t,
t Bohr and Landau D), Littlewood D).
% Bohr and Landau E), Carlson A), Landau A2), Titchmarsh E), Ingham E).

9.16 GENERAL DISTRIBUTION OF ZEROS 231
We shall first prove
Theobem 9.16. If for some X = X(a, T), T1-™ < X < TA,
T
f\fx(s)\*dt =
as T->cc, uniformly for a > a, where l{a) is a positive non-increasing
function with a bounded derivative, and m is a constant ^ 0, then
N(a,T) =
uniformly for a > oc-\-l/logT.
We have /, (,) = «.) T ^ -1 = T ^ ¦
Zw ns ?—1 ns
where a^X) = 0,
aH(X) = % fi(d) = 0 (n<
d\n
and \an(X)\ = ^ pid) ^ d(n)
d\n
d<X
for all n and X.
Let 1-/1 = ?MXB-?MX) = ?(%(*) - h(s)
say, where g(s) = gx(s) and h(s) = ^Y(s) are regular except at s = 1.
Now for a > 2, X > Xo,
so that &(s) 7^ 0. Applying (9.9.1) to &(s), and writing
we obtain
2 T
2t7 J v(a,i-T, T) da = j' {log \h(ao+it)\—log \hB +it)\} dt +
2
-I- f {a,Tgh(a-\-iT) — a,rgh{a+UT)\ da.
J
Now log I*(«)I < log{l + |/A-E)|2} < \fx(s)\2,
so that, if a0 >a,
t t
Jlog|A(a0+»0|d«< ||/a-K
Next

232 GENERAL DISTRIBUTION OF ZEROS Chap. IX
T
so that - f \og\h{2+it)\ dt < ~ =
Also we can apply the lemma of § 9.4 to h{s), with a = 0, fi
m^\, and Mat = 0{XATA). We obtain
for a ^ \. Hence
i
J {argA(a+iT)—argA(or+JiT)}rfo- = O(logX+log T) = 0(log T).
CT0
Hence J v{a, \T, T) da
CT0
Also
2
CTO O'O
if a0 < ax < 2. Taking ax = aQ-\- I/log T, we have
Hence ^(a^ ^T, T) =
Replacing T by \T, \T,... and adding, the result follows.
9.17. The simplest application is
Theobem 9.17. For any fixed a in \ < a < 1,
N(a, T) = 0(T4o<1-or)+c).
We use Theorem 4.11 with x = T, and obtain
m<T n<X
), (9.17.1)
ns
where, if X < T, bn{X) = 0 for n < X and for n > XT; and, as for
an, \bn{X)\ < d{n) = 0(ne). Hence
T T
I Z-4 ns - Z-, n2a ^-> ?4 (mn)a J \mj

9.17 GENERAL DISTRIBUTION OF ZEROS 233
by G.2.1). These terms are of the same order (apart from e's) if
X = T2a-X, and then
T
dt =
s
V °n( X * *
ns
The 0-term in (9.17.1) gives
The result therefore follows from Theorem 9.16.
9.18. The main instrument used in obtaining still better results for
N(a, T) is the convexity theorem for mean values of analytic functions
proved in § 7.8. We require, however, some slight extensions of the
theorem. If the right-hand sides of G.8.1) and G.8.2) are replaced by
finite sums
then the right-hand side of G.8.3) is clearly to be replaced by
In one of the applications a term Ta\og*T occurs in the data instead
of the above Ta. This produces the same change in the result. The only
change in the proof is that, instead of the term
00
r Ma+2a-\_,,,... k
o
P~2u fJv —
we obtain a term
00
J (s) log4a
o
_ ?/w\«+2a-1
~J w
0
Theobem 9.18. If ?(?+i«) = O^log6'*), where c' < f, then
uniformly for \ ^ a ^ 1.
If 0 <S < 1,
T T
Xin*^ 0
2 y y ax(m)ax(n) sm(Tlogm/n)
? ? ml+8n1+8 logm/n
d2{n) sr y d(m)d(n)
2^ Z Z
Z
A'<m<n

234 GENERAL DISTRIBUTION OF ZEROS Chap. IX
Nowf T d2(n) < Axlog3x, > > T; , < ^zlog3z
n<x *—i *—i (mn)* log nm
m<H<x o /
Hence «, «,
= fi±^ V
00
<
X
r (i+t)Aiog>x . A(i+i/f) r
J S3 "*- Xf J
(putting a; = Xytf) < ^(^g X + ~X-
Hence jd^
since Z2S = e2S1°e* > iBSlogZK.
Also, since 1 < logA+A < logA+A~2
for A > 1,
d(m)d(n) ^ ^ d(m)d(n) sr> ^r d(m)d(n)
t log n/m ?-,?-, (mnI+t ?~i ?-> mtn1+t(mnJ log n/m
J!*L f±±
(mn)? log njm J xi
f ^122/ yn)/
I a;2+? A Zw (mn)~z\ogn m
Km<n
1
T
r
Hence
f |/^(l+S+iO|2 ^ < j/^+lW^ (9.18.1)
o
For a = ^ we use the inequalities
dt
n ?-4 ?-* (mn)^logn m
n<X
_j_2 / / j-
n Z-> Z-t (mn)* logn m
7%<iX mains' ~V
< A(T+X)logX,
by G.2.1).
t The first result follows easily from G.16.3); for the second, see Ingham A); the
argument of § 7.21, and the first result, give an extra logx.

9.18 GENERAL DISTRIBUTION OF ZEROS 235
T
Hence J \fx{\+U)\* dt < AT2c(T+X)log2c'(T+2)logX. (9.18.2)
o
The convexity theorem therefore gives
T
*+ 1 8-4
IT
Taking 8 = l/log(T+Z), we obtain
If X = T, the result follows from Theorem 9.16.
For example, by Theorem 5.5 we may take c = J, c' = f. Hence
N(a, T) = ^(Tfd-^log5^). (9.18.3)
This is an improvement on Theorem 9.17 if a > f.
On the unproved Lindelof hypothesis that ?(^-H0 = O(t?), Theorem
9.18 gives
9.19. An improvement on Theorem 9.17 for all values of a in
\ < or < 1 is effected by combining (9.18.3) with
Theorem 9.19(A). N(a, T) = O(T?-°log5T).
We have
T T
\ \fx(h+it)\2 dt <A( \m+it)\2\Mx(l+it)\2 dt+AT
J j
rp
Now M\{s)= V % \cn\^d(n).
Hence
.4
T
j \Mx($+it)\*'d
gg
T
Hence J \fx{%+it)\2 dt < ATl{T+X2)l*log2{T+2)log2X. (9.19.1)

236 GENERAL DISTRIBUTION OF ZEROS Chap. IX
From (9.18.1), (9.19.1), and the convexity theorem, we obtain
T
= o!
(W
If X = Ti, 8 = ljlog(T+2), the result follows as before.
This is an improvement on Theorem 9.17if^<o-<|.
Various results of this type have been obtained,! ^ne most successful J
being
Theorem 9.19 (B). N(a, T) = OG73<1-ffW2-^log5T).
This depends on a two-variable convexity theorem ;§ if
«J(M) = ( f \f(*+it)\^ dt) ,
then J(a,p\+qii) = 0{Jp(<x, A) Jitf, fx)} (a < a <
, 8—a a—a
where p = ? , q = .
p — a p — a
We have
r
J l/x(i+*0l*^ < A ] m+it)fi\Mx(\-\-it)\$ dt+AT
r
< A{TlogHT+2)}l{(T+X)logX}l+AT
< A(T+X)log2(T+X). (9.19.2)
In the two-variable convexity theorem, take a = h, j8 = 1 + 8, A = f,
= h and use (9.18.1) and (9.19.2). We obtain
T
\ \fx(v+it)\VK dt
0
where K = pX+qp lies between \ and |. Taking JC = T, 8 = I/log T,
we obtain r
J ifxiv+WKdt < ATVi-oM-onogfT.
0
f Titchmarsh E), Ingham E), F). % Ingham F). § Gabriel A).

9.19 GENERAL DISTRIBUTION OF ZEROS 237
The result now follows from a modified form of Theorem 9.16, since
A. Selbergf has recently proved
Theorem 9.19 (C). N(a, T) =
uniformly for \ ^ a ^ 1.
This is an improvement on the previous theorem if a is a function of T
such that a—\ is sufficiently small.
9.20. The corresponding problems with a equal or nearly equal to \
are naturally more difficult. Here the most interesting question is that
of the behaviour of x
J N(a,T)da (9.20.1)
i
as T -> oo. If the zeros of ?(s) are j8-Hy> this is equal to
i P
f( 2 l)da= 2 \da=
j3>a,o<y<r j8>J,o<y<r
2 2
Hence an equivalent problem is that of the sum
I IJ3-H. (9.20.2)
There are some immediate results4 If we apply the above argument,
but use Theorem 7.2 (A) instead of Theorem 7.2, we obtain at once
i
f N(a, T) da < ^Tlogfmin/log T,\og-^~)) (9.20.3)
J I \ °o—tl)
for \ ^ a0 ^ 1; and in particular
J N(a, T) da =
= 0{T\og\ogT).
2
These, however, are superseded by the following analysis,
A. Selberg B), the principal result of which is that
J N(a, T)
X
2
We consider the integral
T+U
T
f Selberg E).
da = O(T).
W-H0I-*
| Littlewood D).
(9.20
due
(9.20
•4)
to
.5)

238 GENERAL DISTRIBUTION OF ZEROS Chap. IX
where 0 < U ^ T and if/ is a function to be specified later. We use the
formulae of §4.17. Since
we have Z(t) = z(t)+z(t)+O(t-i), (9.20.6)
where z(t) = (J-VVi*< T n~^il
\27reJ «<x
and x = (</27r)i. Let T
r =
/ / \ —if
Proceeding as in § 7.3, we have
T+U
= O(U2/T)+O(THogT). (9.20.7)
9.21. Lemma 9.21. Let m and n be positive integers, (m, n) = 1,
Jf = max(ra, n).
^ V""/ v ' r<rlMr
The integral is T T r^Ti f l—Ydt.
The terms with m/t = nv contribute
(rn.rmfi
y~L,=u y i __ u y i.
vfi ?-i (rn.rmfi (mn)$ ^-i^r'
m.fi.=nv
The remaining terms are
o
f y y l ) = oiyy
^ ??*? (Pv^\log(nvlmlx) I/ I ^V
og{nv/m(i) \
and the result follows.

9.22 GENERAL DISTRIBUTION OF ZEROS 239
9.22. Lemma 9.22. Defining m, n, M as before, and supposing
<U < T,
T+U
f
"-«'«"" (9.22.1)
if n ^ m. If m < n, the first term on the right-hand side is to be omitted.
The left-hand side is
T+U
U
J \
The integral is of the form considered in § 4.6, with
F{t) = «log-i, c =
Hence by D.6.5), with A2 = (T+U)-1, A3 = (T+U)~2, it is equal to
|logc/T|
with the leading term present only when T ^ c ^ T+U. We therefore
obtain a main term
2 e~27">I/m/ri (9.22.3)
where /a and v also satisfy
^. T'2njm.
The double sum is clearly zero unless n < m, as we now suppose. The
v-summation runs over the range vx ^ v < v2, where vx = T2n/m/x and
v2 = min(T'2n/m^, t), and fi runs over xn/m ^ /z ^ t. The inner sum is
therefore v2 — vl + O(n) if n\n, and O(/i) otherwise. The error term O(n)
contributes O{(mri)%x} = O(MT^) in (9.22.1). On writing fi = nr we are
left with
Let v3 = r'2/mr. Then v2 = v3 unless r < x'2lmx. Hence the error
on

240 GENERAL DISTRIBUTION OF ZEROS Chap. IX
replacing v2 by v3 is
Finally there remains
/m\l
I
n
Now consider the O-terms arising from (9.22.2). The term O(T%) gives
°{Ti 2 2 (^i) =
Next
Suppose, for example, that n < m. Then the terms with r < \nT2jm or
r > 2?iT2/m are
In the other terms, let r — \nT2jrri\—r'. We obtain
°y 2, (nr2l
\r'-e\/(nT2lm)f
omitting the terms r' = — 1, 0, 1; and these are O(Ti+e).
A similar argument applies in the other cases.

9.23 GENERAL DISTRIBUTION OF ZEROS 241
9.23. Lemma 9.23. Let (m, n) = 1 with m,n^ X^ T^.IfT^ ^
then
T+U
T
Let Z@ = z^O + z^O +c@- Then
T+U
T) = O(U2/T)
T
by (9.20.7), and
T+f/
\Z(t)\*dt = O(f/log T) + O(T* + «) = O(U\og T),
T
by Theorem 7.4. Hence
= O{{U*/T)HUlog
T
by Cauchy's inequality. It follows that
T+U
T
T+U
= f {z1
By Lemmas 9.21 and 9.22 the main integral on the right is
T
T+U T+U T+U
\Z(t)e(t)\dt) + l
I \ J /
T 7* T
We have
T+U

242 GENERAL DISTRIBUTION OF ZEROS Chap. IX
whether n^mor not. The result then follows, since
2y + O — ,
\
and since the error terms O{TiX2log(XT)}, O(XT$), O(U2/T), O(TTd)
and O(UXT-l) are all O(f/f T~Hog T).
9.24. Theorem 9.24.
i
J N(a, T) da = 0{T). (9.24.1)
*
Consider the integral
T+U T+U
where ^E) = J Srr1~«
ti(pr)ti(p)H(pr) 2
P<X p<X
Qearly |Sr|<?L
for all values of r. Now
T + U
r<X
J. T
where m = ql(g,r), n = rj(q,r). Using Lemma 9.23, the main term
contributes to this
^ 2
a<xr<x
For a fixed q < X,
1 V V
ZZ
r<X,pr<X
Now (g,r) = 2 4>(v) = 2
K) v\q,v\r

9.24 GENERAL DISTRIBUTION OF ZEROS 243
Hence the second factor on the right is
yy 6bMp} y ^) = y ^ yy
Z<Z 4>(Pr) ?*JK) ^9K) ZZ
<Z 4>(Pr) ?*J ^ ZZ
r<X,pr<X rvr ' v\q,v\r v\q r<X,pr<X
v\r
Put pr = I. Then pv\pr, pv\l, i.e. p\(l/v). Hence we get
v\q KX ^v ' p\(llv)
v\l
The /3-sum is 0 unless I = v, when it is 1. Hence we get
I 0 (q > 1).
Hence
2
and
q<Xr<X {^<f,(p)
Let 4>a(n) be defined by
V 4>a(n) =
71=1
=
ns Us)
so
that
I 1 I pl+a
p\n \ -r
m
m\n p\n
Let tff(n) be defined by
n = l
oo
Then —?{*—1) = ^(«)
71 = 1
and hence nlogn = ^
din
Hence (g, r)log(g, r) =
and 2 2 8,8r(<7,r)log(?,r) = J 0(d) 2
2 ,8r(<7,r)log(?,r) = J 0(d) 2
r<A'
q<X,r<X
2
d<X

244 GENERAL DISTRIBUTION OF ZEROS Chap. IX
Now 0(n) = \lja{n)} = 4>(n
a =
{n) < <j>(n)(logn
Also
pq<X
d\q
4
d\n
Hence 2 18, 8r(g, rJlogfo, r) < 2 log zf V
Since
f
we have V ^ ^ 4 log X.
P<x rvr/
The contribution of all the above terms to / is therefore
on taking, say, X = TtIo.
The O-term in Lemma 9.23 gives
it
Taking say U = Tit, this is O(f/). Hence / = O(U).
By an argument similar to that of § 9.16, it follows that
i
J {N(a, T+U)-N(a, T)} da = 0(U).

9.24 GENERAL DISTRIBUTION OF ZEROS 245
Replacing T by T+U T+2U,... and adding, 0{TJU) terms, we
obtain j
J {N(a, 2T)~N(a, T)} da = 0{T).
k
Replacing T by \T, IT,... and adding, the theorem follows.
It also follows that, if \ < a ^ 1,
a
N(a, T) = -A- f N(o', T) da'
1
id V
da = °(-^-r\ (9.24.2)
h
Lastly, if <j>{t) is positive and increases to infinity with t, all but an
infinitesimal proportion of the zeros of ?(s) in the upper half-plane lie in
the region
The curved boundary of the region
a =
logt
lies to the right of a = ax = | + y^—^
log J
Tl<t<T
and
Hence the number of zeros outside the region specified is o(T\ogT),
and the result follows.
NOTES FOR CHAPTER 9
9.25. The mean value of S(t) has been investigated by Selberg E). One
has
T
j |^ (9.25.1)
for every positive integer k. Selberg's earlier conditional treatment D) is
discussed in §§ 14.20-24, the key feature used in E) to deal with zeros off
the critical line being the estimate given in Theorem 9.19(C). Selberg E)
also gave an unconditional proof of Theorem 14.19, which had pre-
previously been established on the Riemann hypothesis by Little wood.

246 GENERAL DISTRIBUTION OF ZEROS Chap. IX
These results have been investigated further by Fujii [1 ], [2] and Ghosh
[1], [2], who give results which are uniform in k.
It follows in particular from Fujii [1] that
(9.25.2)
and
T
\\S(t + h)-S(t)\^dt<T{AknogC + h\ogT)}k (9.25.3)
o
uniformly for 0 ^ h ^ ^ T. One may readily deduce that
where Nj(T) denotes the number of zeros j5 4- iy of multiplicity exactly j,
in the range 0 < y ^ T. Moreover one finds that
uniformly for X ^ 2, whence, in particular,
0 < yn « T
for any fixed k ^ 0. Fujii [2] also states that there exist constants X > 1
and fi < 1 such that
i~y * (9.25.5)
2n/\ogyn
and
each hold for a positive proportion of n (i.e. the number of n for which
0 < yn ^ T is at least AN(T) if T ^ To). Note that 2yc/log yn is the
average spacing between zeros. The possibility of results such as (9.25.5)
and (9.25.6) was first observed by Selberg [6].
9.26. Since the deduction of the results (9.25.5) and (9.25.6) is not
obvious, we give a sketch. If M is a sufficiently large integer constant,

9.26 GENERAL DISTRIBUTION OF ZEROS
then (9.25.2) and (9.25.3) yield
2T
and
h)-S(t)\4dt<$T
uniformly for
2nM , 4nM
log T log T
By Holder's inequality we have
2T 2T
2T
247
h)-S(t)\*dt
so that
2T
h)-S(t)\dt>T.
We now observe that
S(t + h)-S(t) = N(t + h)-N(t)~
2n ~+°Ug~
for
, whence
2T
h)-N(t)-
hlogT
2n
dt> T.
We proceed to write h = 2nMA./log T and

248 GENERAL DISTRIBUTION OF ZEROS Chap. IX
so that
»rt-«.»-iy-%<.+5f.i
Thus
2T+2nmX/logT
[ \S(t,X)\dt
\
m = 0 J
T+2nmX/logT
2T
= m[ W,X)\dt +
T
and hence
2T
\S(t,X)\dt>T (9.26.1)
T
uniformly for 1 ^ X ^ 2, since M is constant.
Now, if / is the subset of [T, 2T] on which Nit + ^—] = N(t), then
V log TJ
S(t,X) + 2X-2 (te[T,2T]-I),
so that (9.26.1) yields
2T
T< S(t,X)dt + BX-2)T + 2m(I),
T
where m(I) is the measure of /. However
2T
\S(t,X)dt = C T
\logTf
T
whence m(I) > T, if X > 1 is chosen sufficiently close to 1. Thus, if
then
neS

9.26 GENERAL DISTRIBUTION OF ZEROS 249
so that
UeS
by (9.25.4) with k = 2. It follows that
#S>N{T), (9.26.2)
proving that (9.25.5) holds for a positive proportion of n.
Now suppose that \i is a constant in the range 0 < /i < 1, and put
and
y= reU:yn+1~yn^iogTy
T
whence #?7 = —log T+ O(T). Then
Zn
net/
S (yn+i-
net/- V
If the implied constant in (9.26.2) is rj, it follows that # Vp N(T), on
taking /i = 1 - v, with 0 < v < r]{X -1)/A - ?/). Thus (9.25.6) also holds for
a positive proportion of n.
9.27. Ghosh [1] was able to sharpen the result of Selberg mentioned
at the end of §9.10, to show that S(t) has at least
AioeiogT ^
(logloglog D*
-s
sign changes in the range 0 < t < T, for any positive S, and A = A(8),
T ^ T(^). He also proved (Ghosh [2]) that the asymptotic formula
(9.25.1) holds for any positive real k, with the constant on the right hand

250 GENERAL DISTRIBUTION OF ZEROS Chap. IX
side replaced by FBk + 1)/F(k + l)BnJk. Moreover he showed (Ghosh
[2]) that |S@|
say, has a limiting distribution
1 f «2*2dz,
in the sense that, for any a > 0, the measure of the set of t e [0, T ] for
which /(*) ^ a, is asymptotically TP{a). (A minor error in Ghosh's
statement of the result has been corrected here.)
9.28. A great deal of work has been done on the 'zero-density
estimates' of §§9.15-19, using an idea which originates with Halasz [1],
However it is not possible to combine this with the method of §9.16,
based on Littlewood's formula (9.9.1). Instead one argues as follows
(Montgomery [1; Chapter 12]). Let
so that an = 0 for 2 ^ n ^ X. If C(p) = 0, where p = fi + iy and fi > ?, then
we have
e~VY+ Z ann->e-nlY= f
n > X
2 + ioo
= 2_ f
2ni J
2-ioo
by the lemma of §7.9. On moving the line of integration to R(s) = \ —
this yields
OO
f
_ 00
since the pole of F(s) at s = 0 is cancelled by the zero of C(s + p). If we
now assume that log2T ^ y ^ T, and that log T < logX, log Y < log T,

9.28 GENERAL DISTRIBUTION OF ZEROS
then e~1/y> 1 and
¦ u.0 \ Mx{l)T A -p)Y1-" = o(l),
3" /"^whence either
Y a n~pe-n'Y
251
n > X
or
In the latter case one has
for some tp in the range \tp — y| ^ log2T. The problem therefore reduces
to that of counting discrete points at which one of the Dirichlet series
Tann-se-n/Y, Mx(s), and ?(s) is large. In practice it is more convenient to
take finite Dirichlet polynomials approximating to these.
The methods given in §§9.17-19 correspond to the use of a mean-value
bound. Thus Montgomery [1; Chapter 7] showed that
r=l
N
Sa n
n
n = 1
N
for any points sr satisfying
(9.28.1)
11 ^ T, I(sr+ 1-sr)^l, (9.28.2)
and any complex an. Theorems 9.17, 9.18, 9.19(A), and 9.19(B) may all
be recovered from this (except possibly for worse powers of log T).
However one may also use Halasz's lemma. One simple form of this
(Montgomery [1; Theorem 8.2]) gives
R
s
r = 1
N
s
n = 1
N
(9.28.3)
n = 1
for any points sr satisfying (9.28.2). Under suitable circumstances
this implies a sharper bound for R than does (9.28.1). Under the Lindelof
hypothesis one may replace the term RThn (9.28.3) by RTeN$, which is
superior, since one invariably takes N ^ T in applying the Halasz
lemma. (If N^T it would be better to use (9.28.1).) Moreover
Montgomery [1; Chapter 9] makes the conjecture (the Large Values

252 GENERAL DISTRIBUTION OF ZEROS Chap. IX
Conjecture):
R
r = 1
N
Z °nn~°r
n = 1
2 N
<(N+RTe) ? \an\2n-2°
n\
n = 1
for points sr satisfying (9.28.2). Using the Halasz lemma with the
Lindelof hypothesis one obtains
N{a, T)<Te, | + e ^ a ^ 1, (9.28.4)
(Halasz and Turan [1], Montgomery [1; Theorem 12.3]). If the Large
Values Conjecture is true then the Lindelof hypothesis gives the wider
range % + e ^ a ^ 1 for (9.28.4)
9.29. The picture for unconditional estimates is more complex. At
present it seems that the Halasz method is only useful for a ^ %. Thus
Ingham's result, Theorem 9.19(B), is still the best known for ^ < a
Using (9.28.3), Montgomery [1; Theorem 12.1] showed that
N{a, T) < T^-^aog TI4 (f^ a ^ 1),
which is superior to Theorem 9.19(B). This was improved by Huxley [1]
to give
iV(a,T)«T3<i-)/C--i)(logTL4 (}^<x^l). (9.29.1)
Huxley used the Halasz lemma in the form
R<\nV~2 X \an\2n-2° + TNV-6( ? \an\2n-2" }}
for points sr satisfying (9.28.2) and the condition
N
Z ann
— s.
V.
n = 1
In conjunction with Theorem 9.19(B), Huxley's result yields
(c.f. (9.18.3)). A considerable number of other estimates have been given,
for which the interested reader is referred to Ivic [3; Chapter 11]. We
mention only a few of the most significant. Ivic [2] showed that
\ T<9 9">/(8<r-2) +e
which supersede Huxley's result (9.29.1) throughout the range
|- < a < 1. Jutila [1] gave a more powerful, but more complicated, result,

9.29 GENERAL DISTRIBUTION OF ZEROS 253
which has a similar effect. His bounds also imply the 'Density
hypothesis' N(a,T)< T2~2a + e, for ^- ^ a ^ 1. Heath-Brown [6] im-
improved this by giving
N(<r,T)< T<9-9<r)/<7<r-i) + « (ft^o- < 1).
When a is very close to 1 one can use the Vinogradov-Korobov
exponential sum estimates, as described in Chapter 6. These lead to
for suitable numerical constants A and A', (see Montgomery [1;
Corollary 12.5], who gives A = 1334, after correction of a numerical
error).
Selberg's estimate given in Theorem 9.19(C) has been improved by
Jutila [2] to give
N((t,T)< Ti-u-*><»-*> log T
uniformly for ^ ^ a ^ 1, for any fixed S > 0.
9.30. Of course Theorem 19.24 is an immediate consequence of
Theorem 19.9(C), but the proof is a little easier. The coefficients Sr used
in §9.24 are essentially
T logX/r
log A
and indeed a more careful analysis yields
T
logX
logxr
0
Here one can take X^ T^~e using fairly standard techniques, or
X^T&~e by employing estimates for Kloosterman sums (see
Balasubramanian, Conrey and Heath-Brown [1]). The latter result
yields (9.24.1) with the implied constant 0-0845.

X
THE ZEROS ON THE CRITICAL LINE
10.1. General discussion. The memoir in which Riemann first con-
considered the zeta-function has become famous for the number of ideas
it contains which have since proved fruitful, and it is by no means
certain that these are even now exhausted. The analysis which precedes
his observations on the zeros is particularly interesting. He obtains, as
in § 2.6, the formula m
n-H(s) = -7-V,+ U(x)(xh-i+x-l-b) dx,
s{S—l) J
where ip(x) = 2
71=1
Multiplying by ^5E—1), and putting s = \-\-it, we obtain
00
S(*) = i-(t2+l) J t(x)x-$cos{$tlogx) dx. A0.1.1)
1
Integrating by parts, and using the relation
which follows at once from B.6.3), we obtain
a(t) = 4 — {x%ijj'(x)}x~\cos(\t\ogx) dx. A0.1.2)
J dx
1
Riemann then observes:
'Diese Function ist fur alle endlichen Werthe von t endlich, und lasst sich nach
Potenzen von tt in eine sehr schnell convergirende Reihe entwickeln. Da fur einen
Werth von s, dessen reeller Bestandtheil grosser als 1 ist, log ?(s) = — 2 log( 1 —p~s)
endlich bleibt, und von den Logarithmen der iibrigen Factoren von E(t) dasselbe
gilt, so kann die Function E(t) nur verschwinden, wenn der imaginare Theil von
t zwischen %i und — ?i liegt. Die Anzahl der Wurzeln von E(t) = 0, deren reeller
Theil zwischen 0 und T liegt, ist etwa
denn dass Integral J d log E(?) positive um den Inbegriff der Werthe von t
erstreckt, deren imaginare Theil zwischen \i und — \i, und deren reeller Theil
zwischen 0 und T liegt, ist (bis auf einen Bruchtheil von der Ordnung der Grosse
I IT) gleich {Tlog(TJ27r)—T}i; dieses Integral aber ist gleich der Anzahl der in
diesem Gebiet liegenden Wurzeln von E(t) = 0, multiplicirt mit 2-rri. Man findet
nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist
sehr wahrscheinlich, dass alle Wurzeln reelle sind.'

10.1 ZEROS ON THE CRITICAL LINE 255
This statement, that all the zeros of E(t) are real, is the famous
'Riemann hypothesis', which remains unproved to this day. The memoir
goes on:
'Hiervon ware allerdings ein stronger Beweis zu wiinschen; ich habe indess die
Aufsuchung desselben nach einigen fliichtigen vergeblichen Versuchen vorlaufig
bei Seite gelassen, da er fur den nachsten Zweck meiner Untersuchung [i.e. the
explicit formula for tt{x)] entbehrlich schien.'
In the approximate formula for N(T), Riemann's l/T may be a
mistake for log T; for, since N(T) has an infinity of discontinuities at
least equal to 1, the remainder cannot tend to zero. With this correction,
Riemann's first statement is Theorem 9.4, which was proved by von
Mangoldt many years later.
Riemann's second statement, on the real zeros of S(?), is more obscure,
and his exact meaning cannot now be known. It is, however, possible
that anyone encountering the subject for the first time might argue as
follows. We can write A0.1.2) in the form
CO
E(t) = 2 (<&(u)cosutdu, A0.1.3)
where ®(u) = 2 f ('ZnWe*1'—3n27re^u)e-n27relu. A0.1.4)
71-1
This series converges very rapidly, and one might suppose that an
approximation to the truth could be obtained by replacing it by its first
term; or perhaps better by
since this, like ®(u), is an even function of u, which is asymptotically
equivalent to ®{u). We should thus replace E(t) by
oo
E*{t) = 4tt2 j cosh Pwe-27rcosh2lf cos utdu.
o
The asymptotic behaviour of E*(t) can be found by the method of
steepest descents. To avoid the calculation we shall quote known
Bessel-function formulae. We havef
CO
Kz{a) — [ e-acosh "coshz-M du,
and hence a*(t) = ^
For fixed z, as v -> oo
| Watson, Theory of Bessel Functions, 6.22 E).

256 ZEROS ON THE CRITICAL LINE Chap. X
Hence
~V2C
\ r \ i
(t \
\t log (- |-7T
/
The right-hand side has zeros at
t
and the number of these in the interval @, T) is
T T T
jLlog.^——+OA).
The similarity to the formula for N(T) is indeed striking.
However, if we try to work on this suggestion, difficulties at once
appear. We can write
E(t)-E*(t)= | {d>{u)-O*{u)}eiutdu.
To show that this is small compared with E(t) we should want to move
the line of integration into the upper half-plane, at least as far as
l(u) = \tt\ and this is just where the series for O(w) ceases to converge.
Actually |
and |?(-2-H0i is unbounded, so that the suggestion that E*(t) is an
approximation to E(t) is false, at any rate if it is taken in the most
obvious sense.
10.2. Although every attempt to prove the Kiemann hypothesis, that
all the complex zeros of ?(s) lie on a — \, has failed, it is known that
?(s) has an infinity of zeros on a = ?. This was first proved by Hardy
in 1914. We shall give here a number of different proofs of this theorem.
First method.^ We have
E(t) = _i(?+i)W-i*T(i+i»0?(i+*0,
where E(t) is an even integral function of t, and is real for real t. A zero
t Hardy A).

10.2 ZEROS ON THE CRITICAL LINE 257
of ?(s) on<r=| therefore corresponds to a real zero of E(t), and it is
a question of proving that E(t) has an infinity of real zeros.
Putting x = —ia. in B.16.2), we have
00
- —-jcoshoddt = e-^iCL—2e^iocifj{e2iOL)
nit+i A0.2.1)
= 2 cos la— 2e%i(x{\-\-ifj(e2ia)}.
Since ?($+*7) = 0{tA), E{t) = 0{tAe-^), and the above integral may
be differentiated with respect to a any number of times provided that
a < \it. Thus
00
2 f S
J |gp.
7T
0
We next prove that the last term tends to 0 as a -> \tt, for every fixed n.
The equation B.6.3) gives at once the functional equation
_i 1 1 -1 /1\
\xl'
or
Hence
It is easily seen from this that ^+<A(a;) and all its derivatives tend to
zero as x -> i along any route in an angle [arg(x—i)\ < \n.
We have thus proved that
t1^^- A0.2.2)
a-4-7r J t2Jrl
4 j/
Suppose now that E(t) were ultimately of one sign, say, for example,
positive for t > T. Then
lim f
<2ncosha^^ =*L,
say. Hence -^~ t2n
J ' T4
cosh a< ^

258 ZEROS ON THE CRITICAL LINE Chap. X
for all a < \it and T' > T. Hence, making a ->¦ ?tt,
t ^
^ t2n cosh \Trt dt ^ L.
T
Hence the integral ^ ' t2n cosh \nt dt
J t 4
0
is convergent. The integral on the left of A0.2.2) is therefore uniformly
convergent with respect to a for 0 < a < ^n, and it follows that
for every n.
This, however, is impossible; for, taking n odd, the right-hand side
is negative, and hence
oo T
f J^) fin cosh 1 nt dt<- [ -^ t^ COsh |-7r< rf<
o
where K is independent of w. But by hjrpothesis there is a positive
m = m(T) such that Z(t)/(t2+l) > m for 2T < < < 2T+1. Hence
oo 2T+1
f ^!?i t*n cosh j^ dt^[ mt2n dt > m{2TJn.
Hence m22" < iT,
which is false for sufficiently large n. This proves the theorem.
10.3. A variant of the above proof depends on the following theorem
of Fejerif
Let n be any positive integer. Then the number of changes in sign in
the interval @, a) of a continuous function f(x) is not less than the number
of changes in sign of the sequence
a a
/@), jf(t)dt, ..., jf(t)t"dt. A0.3.1)
o o
We deduce this from the following theorem of Fekete.-J
t Fejer A). % Fekete A).

10.3 ZEROS ON THE CRITICAL LINE 259
The number of changes in sign in the interval @, a) of a continuous
function f(x) is not less than the number of changes in sign of the sequence
/(a), Ma), ..., fn(a), A0.3.2)
where x
/» = J A-i@ dt (" = 1» 2,..., »), /o(x) = /(*).
o
To prove Fekete's theorem, suppose first that n = 1. Consider the
curve y = /xfc). Now /x@) = 0, and, if f(a) and /x(a) have opposite
signs, y is positive decreasing or negative increasing at x = a. Hence
f(x) has at least one zero.
Now assume the theorem for n — 1. Suppose that there are k changes
of sign in the sequence Mx),..., fn(x). Then fx{x) has at least k changes
of sign. We have then to prove that
(i) if f(a) and Ma) have the same sign, f(x) has at least k changes of
sign,
(ii) iff {a) and Ma) have opposite signs, f{x) has at least k-\-l changes
of sign.
Each of these cases is easily verified by considering the curve y =Mx).
This proves Fekete's theorem.
To deduce FejeVs theorem, we have
X
1 r
Jv\) ~*"~~ "# \~T I \*^" '/ J\ ) ^*'j
0
and hence
/» = r-nrr f (°-')y-V@ dt = 7-^m f fa-W dt-
(v—iy.j {v—iy.j
0 0
We may therefore replace the sequence A0.3.2) by the sequence
a a
f(a), jf(a-t)dt, ..., J fia-t)^-1 dt. A0.3.3)
j J
o o
Since the number of changes of sign off(t) is the same as the number
of changes of sign oif(a—t), we can replace f(t) by f(a—t). This proves
FejeVs theorem.
To prove that there are an infinity of zeros of ?(s) on the critical line, we
prove as before that
f —l)n7TCOSi7T
a
Hence f -—3- f-n cosh at dt
0

260 ZEROS ON THE CRITICAL LINE Chap. X
has the same sign as ( — l)n for n — 0, 1,..., N, if a = a(N) is large
enough and a = a(N) is near enough to \n. Hence E(t) has at least N
changes of sign in @, a), and the result follows.f
10.4. Another methodJ is based on Riemann's formula A0.1.2).
Putting x = e2u in A0.1.2), we have
CO
S@ = 4 f — {e*u4f'(e2u)}e-$u cos ut du
J du
o
CO
= 2 J <b(u)coaut du,
o
say. Then, by Fourier's integral theorem,
CO
1 r
(S?(u) = - E(t)cosut dt,
TT J
0
and hence also
CO
f
J
E{t)t2ncos ut dt.
Since ip(x) is regular for R(a;) > 0, <E>(w) is regular for — \-n < \{u) < \tt.
Let
<b(i) c+2+w4+... (\u\ <
Then Bn)!cn = ( —l)^2n)@) = - f E{t)t2n dt.
o
Suppose now that E(t) is of one sign, say E(t) > 0, for t > T. Then
cn > 0 for n > n0, since
co T+2 T
fS@<2nd<> J E{t)t2ndt— \ \E{t)\t2n dt
o r+i b
T+2 T
> (T+lJn [ E{t) dt-T2n J |S(<)| dt.
T+l 0
It follows that <fr(n)(iu) increases steadily with u if n > 2?i0. But in fact
O(w) and all its derivatives tend to 0 as u -> ^V along the imaginary
axis, by the properties of ip(x) obtained in § 10.2. The theorem therefore
follows again.
10.5. The above proofs of Hardy's theorem are all similar in that
they depend on the consideration of 'moments' J f(t)tn dt. The following
t Fekete B). J Polya C).

10.5 ZEROS ON THE CRITICAL LINE 261
methodf depends on a contrast between the asymptotic behaviour of
the integrals 2t ht
j Z{t)dt, J \Z(t)\dt,
T T
where Z{t) is the function defined in § 4.17. If Z(t) were ultimately of
one sign, these integrals would be ultimately equal, apart possibly from
sign. But we shall see that in fact they behave quite differently.
Consider the integral
J {x(«)}-*?(«) da,
where the integrand is the function which reduces to Z(t) on a = \,
taken round the rectangle with sides a = \, a = f, t = T, t = 2 T.
This integral is zero, by Cauchy's theorem. Now
\+2iT IT
j &(«)}-*?(«) da = i | Z(t) dt.
1+ IT T
Also by D.12.3)
Hence, by E.1.2) and E.1.4),
ii +<) (i < or < 1),
1< a < 1).
The integrals along the sides t = T, t = 2T are therefore O(yf+e).
The integral along the right-hand side is
2T
The contribution of the 0-term is
J 0{rt)
T
The other term is a constant multiple of
2T
2T
71 == 1 «?
Now ^(jaog|««log»)
Hence, by Lemma 4.5, the integral in the above sum is 0{Tl), uniformly
with respect to n, so that the whole sum is also ?
f See Landau, Vorlesungen, ii. 78-85.

262 ZEROS ON THE CRITICAL LINE
Combining all these results, we obtain
2T
$ Z{t)dt= 0{Tl).
T
On the other hand,
2T 2T
\\z{t)\dt= \ m+it)\
T T
But
2 + iT 2 + 2tT
r + 1
Chap. X
A0.5.1)
H) dt
2T
T
?2»'log
2+iT 2+*2iT
J
Hence
27*
J \Z{t)\ dt > AT.
T
A0.5.2)
Hardy's theorem now follows from A0.5.1) and A0.5.2).
Another variant of this method is obtained by starting again from
A0.2.1). Putting a = \n—8, we obtain
CO
f
J
= O(l)+O{ I exp(-nVie-««)}
f e-»iff
f
as 8 -> 0. If, for example, E(t) > 0 for t > <0, it follows that for T > t0
2T
f
2T
f Z{t) dt < A f -|^L<i
2r
J
l«T A
to
This is inconsistent with A0.5.2), so that the theorem again follows.
10.6. Still another methodf depends on the formula D.17.4), viz.
Z(t) = 2
f Titchmarsh A1).

10.6 ZEROS ON THE CRITICAL LINE 263
where x = %/(</2tt). Here & = &(t) is defined by
so that
2 r(i-ia) 2(i+
CO
I I
^-R J
0
and we have q,/-a ¦, -, . , * o < n,-. U\
& (t) = ? log t—\ log 27r-i- 0A/0,
V cos(<ylogn) T
= 2 —vk— = 1
The function #(?) is steadily increasing for t ^ <0. If ^ is any positive
integer (^ v0), the equation i?(<) = vn therefore has just one solution,
say tv, and tv ~ 27rv/log v. Now
The sum
consists of the constant term unity and oscillatory terms; and the
formula suggests that g(tv) will usually be positive, and hence that Z(t)
will usually change sign in the interval (tv, tv+1).
We shall prove
Theorem 10.6. As N -+ oo
f Z(t2v) ~ 2N, f Z(t2v+1) ~ -2N.
V=Vo V=Vo
It follows at once that Z(t2v) is positive for an infinity of values of v,
and that Z{t2v+1) is negative for an infinity of values of v\ and the
existence of an infinity of real zeros of Z(t), and so ofE(t), again follows.
We have
V nit \ V V
Z g{f2v)- Z Z
v = M+l v=M+l n^«/
= N—M+
where t = max(^2ij,/+2J 2ttti2). The inner sum is of the form

264 ZEROS ON THE CRITICAL LINE Chap. X
where <f>(v) = h
We may define ty for all v ^ v0 (not necessarily integral) by #(?,,)
Then
so that 0» =
Hence ^'(v) is positive and steadily decreasing, and, \iv is large enough,
< ^ ^ .
^ t2Xlog%N
Hence, by Theorem 5.9,
2 cosfe, log •) = o Ls
Hence
Hence f
and a similar argument applies to the other sum.
10.7. We denote by N0(T) the number of zeros of ?(s) of the form
\-\-it @ < t < T). The theorem already proved shows that N0(T) tends
to infinity with T. We can, however, prove much more than this.
Theorem 10.7.t N0(T) >AT.
Any of the above proofs can be put in a more precise form so as to
give results in this direction. The most successful method is similar in
principle to that of § 10.5, but is more elaborate. We contrast the
behaviour of the integrals
t + H x t + H x
where T < t < 2T and T -> oo.
f Hardy and Littlewood C).

10.7
ZEROS ON THE CRITICAL LINE
265
We use the theory of Fourier transforms. Let F(u), f(y) be functions
related by the Fourier formulae
00
00
F{u) =
—
Integrating over (t,t-\-H), we obtain
'v* dy, f(y) = * J F(u)e~^ du.
t+H
oo
r i r pivH i
F(u) du = 17— f(y) e—r-± e** dy,
t + TI
so that
F(u)du, f(y)—.
ly
are Fourier transforms. Hence the Parseval formula gives
oo t + H
II"
u)du
dt =
CO
J
— 00
dy.
If F(u) is real, \f(y) | is even, and we have
oo t+TI
r r F(u)du
2 oo
dt — 2 I ' f("Al2
o
4 sin2 \Hy
oc
J
Now B.16.2) may be written
L
— 00
Putting ? = — id77"—P)—y, it is seen that we may take
F(t) =
A0.7.1)
Let # > 1. The contribution of the first term in f(y) to A0.7.1) is clearly
0(H). Putting y = logic, G = ellH, we therefore obtain
t + H
J J F(u)
du
( °
= 0{H2 f IMe^-tW
I i
log2*
. A0.7.2)

266 ZEROS ON THE CRITICAL LINE Chap. X
Now
g-n2nxHs'm8+ico88)
71=1
2
00
__ y e-2n»7TX28inS i y y e-{7n2tHi2OTX2sinS+i(m2-7i2OTX2cosS
71=1 m^n
As in § 10.5, the first sum is Ofa^S-i), and its contribution to A0.7.2) is
therefore
o
f *^
= O{H2(G-l)8-l}+O(S-lllogG) = O(HB-i).
The sum with m ^ n contributes to the second term in A0.7.2) terms
of the form
J
JZ _ Q\1
J log2* \\m2-n2\Glog2G)
(jy2e-{m2+ni!OT sin S\
|m2-7l2| )
by Lemma 4.3. Hence the sum is
m ?-t m—nl
n = l ' x m = 2 71 = 1 '
=
m
= o/^r2 log2 i] =
\ /
for 8 < ho(H). The first integral in A0.7.2) may be dealt with in the
same way. Hence
oo
-00
t + H
J J
du
dt =
Taking 5 = 1/T and T > To(#), it follows that
2T
J \I\2dt= 0(HTi). A0.7.3)
10.8. We next prove that
J > (AH+Y)T~i, A0.8.1)
where f |T|2 ^ = 0(T) @ < H < T). A0.8.2)

10.8 ZEROS ON THE CRITICAL LINE
We have, if s = \+it, T < t < 2T,
267
Hence
t + H
>A
l + H
du>A
2 3*
AT
t+H
J
du
=AH+0\L2
O(HT-i).
It is now sufficient to prove that
2T
S 2
dt = O(T),
and the calculations are similar to those of § 7.3, but with an extra
factor log m log n in the denominator.
To prove Theorem 10.7, let S be the sub-set of the interval (T,2T)
where I = J. Then r r
J \I\dt = J J dt.
s
S
2T
2T i 2T
Now J \I\dt^ J \I\dt^ ItJ \I\2dt
by A0.7.3); and by A0.8.1) and A0.8.2)
J Jdt> T-i j (AH+Y) dt
2T
-i
AT-iHm(S)-T-i
dt
-T-$\T J \Y\2dtJ
where m(S) is the measure of S. Hence, for H ^ 1 and T > T0(H),
m(S) < ATH-i.

268 ZEROS ON THE CRITICAL LINE Chap. X
Now divide the interval (T, 2T) into [TJ2H] pairs of abutting intervals
ii> J2> each, except the last j2, of length H, and each j2 lying to the
right of the corresponding jv Then either jx or j2 contains a zero of
E(t) unless j1 consists entirely of points of 8. Suppose that the latter
occurs for v j^s. Then
vH < $
Hence there are, in {T, 2T), at least
() >
zeros if H is large enough. This proves the theorem.
10.9. For many years the above theorem of Hardy and Littlewood,
that N0(T) > AT, was the best that was known in this direction.
Recently it has been proved by A. Selberg B) that N0(T) > AT log T.
This is a remarkable improvement, since it shows that a finite propor-
proportion of the zeros of ?(s) lie on the critical line. On the Riemann hypo-
hypothesis, of course,
Nc(T) = N(T)~±-TlogT.
Ztt
The numerical value of the constant A in Selberg's theorem is very
small. |
The essential idea of Selberg's proof is to modify the series for ?(s)
by multiplying it by the square of a partial sum of the series for {?(s)}~?.
To this extent, it is similar to the proofs given in Chapter IX of theorems
about the general distribution of the zeros.
We define <xv by
It is seen from the Euler product that c^ <xv = a^v if (fi, v) = 1. Since
the series for A—z)i is majorized by that for A—z)~i, we see that, if
v=l
then \<xv\ < 0^ < 1.
Let A = a,(
Then \p,\ < 1.
f It was calculated in an Oxford dissertation by S. H. Min.

10.9 ZEROS ON THE CRITICAL LINE 269
All sums involving /?„ run over [1, X] (or we may suppose fiv = 0 for
v > X). Let
10.10. Lett
c+ioo
f
J
i
c—ioo
where c > 1. Moving the line of integration to a — \, and evaluating
the residue at 5 = 1, we obtain
— 00
On the other hand,
^ C+tOO
i /
n=l y. v c._ioo
22?(
n=l ft v v
Putting 2 = e~^*n~^~v', it follows that the functions
exp —
n=l u. v v
are Fourier transforms. Hence, as in § 10.7,
oo . t + h i2 I/A oo
J | F(u)du\ dt < 2h* J \f(y)\2dy+8 j \f(y)\2y-2dy A0.10.1)
-oo ^t ' 0 ljh
where h < 1 is to be chosen later.
Putting y = logx, G = ellh, the first integral on the right is equal to
g . ..
n=l fi v
f Titchmarsh B6).
2
dx.

270 ZEROS ON THE CRITICAL LINE Chap. X
Calling the triple sum g(x), this is not greater than
o o a
2 J1^^^^
f f
1 •*
Similarly the second integral in A0.10.1) does not exceed
00
f IS'^) |2 -7
J log2*
10.11. We have to obtain upper bounds for these integrals as 8 -> 0,
but it is more convenient to consider directly the integral
J(x,0) = | \g(u)\2u-edu @ < 0 ^ J, x > 1).
X
This is equal to
j expl-rrkr + ^Wsi
x
//t /C ft
____
m=l n=l kA^
Let Sx denote the sum of those terms in which w/c/A = nfi/v, and
the remainder. Let (kv,X/jl) = q, so that
kv = aq, Xfi = bq, (a,b) = 1.
Then, in 21? ma = nb, so that n = m, m = rb (r = 1,2,...). Hence
i J \ 3 / u
r=1 x
Now
xvt7
The last r-sum is of the form

10.11
ZEROS ON THE CRITICAL LINE
271
where KF), and later K-^d), are bounded functions of 6. Hence we
obtain
y' dy+
+
Putting r) = 2nK2iJL2q~2 sin 8, it follows that
where
Defining <f>a(n) as in § 9.24, we have
Y~'?
I
p\q
4>-e(p)-
Hence
S@) =
Let ?Z and ^x denote positive integers whose prime factors divide p.
Let k = d,K', v = dxv', where (k',p) = 1, (v',p) = 1. Then
f
Now, for (k', p)=l, $dK. = ^ log -* .
Hence the above sum is equal to
1 V "d(*d' V -?El- log — V
g*X Z &-% Z k'1-0 gdK' Z
log
' g
2L
10.12. Lemma 10.12. Tf e
p\P
uniformly with respect to 6.

272 ZEROS ON THE CRITICAL LINE Chap. X
We may suppose that X ^ 2d, since otherwise the lemma is trivial.
l + ioo
Now
Also
-^ f %ds = O @<a;< 1), logo: (x > I).
Ztt% j S
1—100
p\p
Hence the left-hand side of A0.12.1) is equal to
A0.12.2)
There are singularities at 5 = 0 and s = 6. If 6 ^ {log(X/d)}~1, we can
take the line of integration through s — 6, the integral round a small
indentation tending to zero. Now
<A\t\
for all t (large or small). Also
Hence A0.12.2) is
«HMSHf(fn(»
and the result stated follows.
If 6 < (log(X/<i)}-1, we take the same contour as before modified by
a detour round the right-hand side of the circle \s\ = 2{log(X/d)}-1.
On this circle
Alog(Xld).
the ^-product goes as before, and
Hence the integral round the circle is
v p\p v *' " v p\p v Jri '
The integral along the part of the line a = 6 above the circle is
00
The lemma is thus proved in all cases.

10.13 ZEROS ON THE CRITICAL LINE 273
10.13. Lemma 10.13.
Defining <x'd as in § 10.9, we have
Y I *<**&! < V "d<*d* - V -
A ddx ^ A ddx Ad'
p\dd\ p\dd\ p\D
where D is a number of the same class as d or dx,
' pIp
10.14. Lemma 10.14.
uniformly with respect to 6. In particular
By the formulae of § 10.11, and the above lemmas,
p\kp v 6 p\ddi l \ / 1 p\p
logXl 1\ ^ p A ddx
pip' plddi
= 0,
Hence
¦ (
since
Uog2X
n|p

274
Hence
ZEROS ON THE CRITICAL LINE
Chap. X
10.15. Estimation of Sx. From A0.11.1), Lemma 10.14, and
the inequality \f$v\ ^ 1, we obtain
ej
We shall ultimately take X = 8-° and A = (alogX)'1, where a and c
are suitable positive constants. Then G = Xa = 5~ac. If x ^ G, the last
two terms can be omitted in comparison with the first if GX2 = O{5~%),
i.e. if (a + 2)c ^ \. We then have
10.16. Estimation of 22. If P and ^ are positive, and x ^ 1,
e.g. by applying the second mean-value theorem to the real and
imaginary parts. Hence
2K2
• *11
l •
The terms with m/c/A > 7i/u,/v contribute to the m, n sum
0{ > e-
km=l
m2/c2 n2fjL2
Z I A2 " V2 ) )'
Now
and >
j(—, rriKv—
n
frriK
A2 v2 A \ A
A2v
nAfi
+
Afi zAfi

10.16 ZEROS ON THE CRITICAL LINE
Hence the m, n sum is
275
1 + 1
K \ AfJ.
PI
— + — log2
0 K[i 3
since, as in §10.15, we have X— d~c, with 0 < c < ^. The remaining
terms may be treated similarly. Hence
10.17. Lemma 10.17. Under the assumptions of § 10.15
12
r r
dt = O[
h
A0.17.1)
By A0.15.1) and A0.16.1),
uniformly with respect to d. Hence
A0.17.2)
go o
f |g(x)|2 dx=- f xe ~ dx = [~xdJ]° + d f
J
\ _ I log G \
taking, for example, 9 = \. Also
OO
dJ{G, 6) dd = J
O
dx-8 dd
00
G
00
J
2

276 ZEROS ON THE CRITICAL LINE Chap. X
since G = ellh ^ e. Hence
i
r \q(x)\2 -, r
lgv n dx <;
J log2z J
O 0
= ol f dd \\o( x \ ol x \
\ J &G8 log X ^~ \84(?i log X) \S4 log G log XJ'
x o '
Also <^@) = 0(X), <f>(l) = 0(logX). The result therefore foUows from
the formulae of § 10.10.
10.18. So far the integrals considered have involved F(t). We now
turn to the integrals involving \F(t)\. The results about such integrals
are expressed in the following lemmas.
OO
r
Lemma 10.18.
By the Fourier transform formulae, the left-hand side is equal to
oo oo
2
2 j \f(y)\2dy=2 J
i
oo
2x
dx
<4 (
i
Taking x = 1, 9 = {logU/S)} in A0.17.2), we have
00
J
du =
a
r
Hence
We can estimate the integral over (8~2, oo) in a comparatively trivial
manner. As in § 10.11, this is less than
z C exp -
z z z C exp -7rn^+^"

10.18 ZEROS ON THE CRITICAL LINE 277
Using, for example, k2X~2sin3 > AX~23 > A32 (since X—3~c with
, and |jjw|< 1, this is
f f e-^(m2+n2)8*u* dy
m = l n=!gJ2 j
= o(x2log2X J e-^»' du\ = O(X2log2X
which is of the required form.
10.19. Lemma 10.19.
— 00 V t
For the left-hand side does not exceed
oo , t + h » oo u oo
J Ih J \F{u)\2 du\ dt = h J |#(m)|2 du j dt = h2 J
— oo < —oo w — h —oo
and the result follows from the previous lemma.
10.20. Lemma 10.20. 7/8 = 1{T,
T
J \F{t)\dt>ATl.
o
We have
2 + iT
2 + i 2+iT
Since <?(s) = 0(X\) for a > \, the first term is 0(X), and the third is
0{XT\). Also
n=2
where |an| ^ ^3(^). Hence
2 + iT ^ 2+iT
1 / i O 1/T*"I O I /T O " "~~ ^ I / I I I ^^ /f I
2+i n~i 2+i
vn=2
It follows that J Ui+it)<f>2(±+it) dt — T.

278
Hence
ZEROS ON THE CRITICAL LINE
Chap. X
T
J \F(t)\dt>A JH
o o
dt
7'
I
10.21. Lemma 10.21.
J dt J
> AhTl.
u
The left-hand side is equal to
T+h mm(T,u) T
/• r r r r
\F(u)\ du dt ^ \F(u)\ du dt — h \F(u)\ du,
0 max@,it- A) A u—h A
and the result follows from the previous lemma.
10.22. Theorem 10.22.
N0{T) > ATlogT.
Let E be the sub-set of @, T) where
t+h t+h
J \F(u)\du > J F{u)du.
t t
For such values of t, F(u) must change sign in (t,t+h), and hence so
must S(^), and hence U^-j-iu) must have a zero in this interval.
Since the two sides are equal except in E,
t+h ,t+h t+h ^
dt
dt
! dt ! \F(u)\ du^il f \F{u)\ du- f F(u) du
- " E [ t 1
E t
E [ t
T tt+h
T tt+
= J J
0 ^ t
AhTl- J J F(u) du
du~
t+h
1
! F{u) du
dt.
The left-hand side is not greater than
,t+h
aJ ' A \2^1 I °° ft+h \ 2
\F(u)\ du\ dt\2 < lm(E) ! ! \F{u)\ du) dt
t. ' ' * -oo ^ t '

10.22 ZEROS ON THE CRITICAL LINE 279
by Lemma 10.19 with 8 = 1/T. The second term on the right is not
greater than
LT T t+h 2 \1 ' '
[ dt J J F(u)du dty <
0 0 t '
by Lemma 10.17. Hence
> A11 2L „} Ac
where Ax and A2 denote the particular constants which occur. Since
X = Tc and h = (alogX)-1 = (aclog T)-1,
{m{E)}\ > A^Ti-AziacftT*.
Taking a small enough, it follows that
m{E) > A3 T.
Hence, of the intervals @, h), (h,2h),... contained in @, T), at least
[Az T/h] must contain points of E. If (nh, (n-j-l)h) contains a point t of
E, there must be a zero of t,(\-\-iu) in (t, t-j-h), and so in (nh, (n-j-2)h).
Allowing for the fact that each zero might be counted twice in this way,
there must be at least
$AzT/h]>ATlogT
zeros in @, T).
10.23. In this section we return to the function E*(t) mentioned in
§ 10.1. In spite of its deficiencies as an approximation to E(t), it is of
some interest to note that all the zeros of E*(t) are real.\
A still better approximation to <&(u) is
<&**(u) = 7rB7r cosh |^—3 cosh |^)e-27rcosh2u.
OO
This gives E**(t) = 2 j ®**(u)cosut du,
o
and we shall also prove that all the zeros of E**(t) are real.
The function Kz(a) is, for any value of a, an even integral function
of z. We begin by proving that if a is real all its zeros are purely
imaginary.
It is known that w = Kz(a) satisfies the differential equation
— la—\ = (a + -)w
da\ da) \ a)
This is equivalent to the two equations
dw W dW I , z2\
da a da \ a]
| P61ya A), B), D).

280 ZEROS ON THE CRITICAL LINE Chap. X
These give f-(Ww) = -{\W\2+{a2+z2)\w\2}.
It is also easily verified that w and W tend to 0 as a -> oo. It follows
that, if w vanishes for a certain z and a = a0 > 0, then
OO
\ {\W\*+(aZ+z*)\tv\z}~- = 0.
J a
Taking imaginary parts,
00
2ixy [^da = 0.
J a
Here the integral is not 0, and Kz(a) plainly does not vanish for z real,
i.e. y — 0. Hence x — 0, the required result.
We also require the following lemma.
Let c be a positive constant, F(z) an integral function of genus 0 or 1,
which takes real values for real z, and has no complex zeros and at least
one real zero. Then all the zeros of
F(z+ic)+F(z-ic) A0.23.1)
are also real.
00
We have F{z) = Cz«eaz i__U/««,
where C, a, av... are real constants, ocn =? 0 for ti = 1, 2,..., 2 an
is convergent, q a non-negative integer. Let z be a zero of A0.23.1). Then
\F(z+ic)\ = \F(z-ic)\,
so that
, F{z-ic)
If 2/ > 0, every factor on the right is < 1; if y < 0, every factor is > 1.
Hence in fact y = 0.
The theorem that the zeros of H*(?) are all real now follows on taking
F(z) = KlizB7r), c = I
10.24. For the discussion of E**(?) we require the following lemma.
\f(t) | < Ke~lt[i+S for some positive 8, so that
F{z) = jk) i mei*
dt
— 00

10.24 ZEROS ON THE CRITICAL LINE 281
is an integral function of z. Let all the zeros of F(z) be real. Let <f>(t) be
an integral function of t of genus 0 or 1, real for real t. Then the zeros of
00
it)eiet dt
— 00
are also all real.
We have <j>(t) = CPe* FT (l —-W™,
L\ \ a<»)
where the constants are all real, and 2 am2 *s convergent. Let
Then <j>n{t) -> <f>(t) uniformly in any finite interval, and (as in my Theory
of Functions, § 8.25)
uniformly with respect to n. Hence
00
G(z) = Km -J— f f(t)<f>n(it)e™ dt = lim Gn(z),
n->oo \J\Ztt) j n—>-oo
say. It is therefore sufficient to prove that, for every n, the zeros of
On{z) are real.
Now it is easily verified that F(z) is an integral function of order less
than 2. Hence, if its zeros are real, so are those of
{D-a)F{z) = e^—le-^Fiz)}
dz
for any real a. Applying this principle repeatedly, we see that all the
zeros of
H(z)
= I»{D-all)...(D-*n)F{z) = -* f f{t)(itWt-*i).-4it-«n)e** dt
— 00
are real. Since
4
ocv..cxn \ OC-L
the result follows.
Taking f{t) = 4VB7r)e-27rC0Sh2'
we obtain F(z) = K±iz{2ir),
all of whose zeros are real. If

282 ZEROS ON THE CRITICAL LINE Chap. X
then O(z) = S*(z), and it follows again that all the zeros of S*(z) are
real. If / 9 3 5 \
then G(z) = E**(z). Hence all the zeros of H**(z) are real.
10.25. By way of contrast to the Riemann zeta-function we shall
now construct a function which has a similar functional equation, and
for which the analogues of most of the theorems of this chapter are true;
but which has no Euler product, and for which the analogue of the
Riemann hypothesis is false.
We shall use the simplest properties of Diriclilet's Z-functions (mod 5).
These are defined for a > 1 by
71=1
00
ns
+
tis 1* 2*
71=1
71 = 1
Li(s) =
71s Is 2s
71 = 1
Each x(n) has the period 5. It is easily verified that in each case
X{m)x{n) =
if m is prime to n; and hence that
It is also easily seen that
so that ?0E) is regular except for a simple pole at 5 = 1. The other
three series are convergent for any real positive 5, and hence for cr > 0.
Hence L^s), L2(s), and L3(s) are regular for a > 0.
Now consider the function
/(*) = l86cd{e-<9L1{8)+(ii0Lf{8)}
_ 1 tanfl tanfl 1 1
~p"i 2~» 3s 4s 6s
= ^ «(*, i)+tan 6 ?(8, l)-tan
o
where ?(s, a) is defined as in § 2.17.

10.25 ZEROS ON THE CRITICAL LINE 283
B-l7)f(s) is an integral function of s, and for a < 0 it is equal to
X f
f
\ 5
00 /
^S I QlTI | 4- Q T"\ H ni T"|
^—t \ 5 5
4r(l
5sBttI-s ^\ 5 /
If sin^ + tan0sin— = tan 01 sin — -f- tan 6 sin — V A0.25.1)
5 5 \ 5 5 /
this is equal to
The equation A0.25.1) reduces to
. ofl _ 2tt V5-1
sm2^ = 2 cos— = ,
5 2
and we take 6 to be the root of this between 0 and \-n. We obtain
—1
.277,, , . 477 V5
sm f-tanflsm— = —,
5 5 2
and f(s) satisfies the functional equation
77
There is now no difficulty in extending the theorems of this chapter
to f(s). We can write the above equation as
and putting s = %-\-it we obtain an even integral function of t analogous
to E(t).
We conclude that f(s) has an infinity of zeros on the line a = \, and
that the number of such zeros between 0 and T is greater than A T.
On the other hand, we shall now prove that f{s) has an infinity of
zeros in the half-plane a > 1.

284 ZEROS ON THE CRITICAL LINE Chap. X
If p is a prime, we define a(p) by
so that cx(p) = ± 1 or ±i. ?
For composite %, we define a(n) by the equation
Thus \a(n)\ is always 0 or 1. Let
M(s,X) = y ^m = FI (i-a-
where ^ denotes either xi or X2-
Now
«(p)x*(P) = l(
and these are conjugate since ^i = x\ an(^ Xi an(l Xi Xz are rea^ Hence
xi) and M(s, x2) are conjugate for real 5, and N(s) is real.
Let 5 be real, greater than 1, and -> 1. Then
logM{s,Xi) =
p
p p
Now xl = Xz and XiXz = Xo- Hence
p p *
p
P P
Hence
log if (a, xi) = i(l-*)log—+0A),
N(s) = RM{s,Xl) =
It is clear from this formula that N(s) has a zero at each of the points
(m = l} 2,...).

10.25 ZEROS ON THE CRITICAL LINE
Now for a ^ 1 -j-S, and x = Xi or Xv
log L(s-\-ir, x)~log -3/E, x)
\ <x{p)x(P)}
285
- 2 H
p^p
po
= 0
l^b)-^~iTl
Let a(p) = e27™^. By Kronecker's theorem, given q, there is a number
r and integers a7p such that
2
P*5
Then
2tt
\"(p)-p-ir\ =
1
Hence log L(s+ir, x)-\ogM(s, x) =
-±-\
and we can make this as small as we please by choosing first P and
then q. Using this with xi and X'2> it follows that, given e > 0 and
8 > 0, there is a r such that
\f(s+ir)-N(s)\ < 6 (or ^ 1+8).
Let 5j > 1 be a zero of N(s). For any r\ > 0 there exists an ^ with
0 < i?! < 7], 7]x < 5X— 1, such that N(s) ^ 0 for \s—5j| = ^j. Let
e = min \N(s)\
and S < sl—7]1—l. Then, by Rouche's theorem, N(s) and
N(8)-{N(8)~f(8+ir)}
have the same number of zeros inside \s— s^ = 7I} and so at least one.
Hence f(s) has at least one zero inside the circle \s—Sj—ir\ = r)v
A slight extension of the argument shows that the number of zeros of
f(s) in a > 1, 0 < t < T, exceeds AT as T ~> oo. For by the extension
of Dirichlet's theorem (§8.2) the interval {tQ,mqptQ) contains at least m
values of t, differing by at least t0, such that
2tt
¦x.
p
n
The above argument then shows the existence of a zero in the neigh-
neighbourhood of each point 51+*(

286 ZEROS ON THE CRITICAL LINE Chap. X
The method is due to Davenport and Heilbronn A), B); they proved
that a class of functions, of which an example is
Y Y
1
has an infinity of zeros for a > 1. It has been shown by calculation!
that this particular function has a zero in the critical strip, not on the
critical line. The method throws no light on the general question of
the occurrence of zeros of such functions in the critical strip, but not
on the critical line.
NOTES FOR CHAPTER 10
10.26. In § 10.1 Titchmarsh's comment on Riemann's statement about
the approximate formula for N(T) is erroneous. It is clear that Riemann
meant that the relative error {N(T)-L(T)}/N(T) is O(T~l).
10.27. Further work has been done on the problem mentioned at the
end of § 10.25. Davenport and Heilbronn A), B) showed in general that if
Q is any positive definite integral quadratic form of discriminant d, such
that the class number h(d) is greater than 1, then the Epstein Zeta-
function
CQ(s)= ? Q(x,y)
-S
x> y = —
(x,y) ^(
has zeros to the right of a = 1. In fact they showed that the number of
such zeros up to height T is at least of order T (and hence of exact order
T). This result has been extended to the critical strip by Voronin [3],
who proved that, for such functions (Q(s), the number of zeros up to
height T, for \ < ax < I(s) < o2 < 1, is also of order at least T(and hence
of exact order T). This answers the question raised by Titchmarsh at the
end of § 10.25.
10.28. Much the most significant result on N0(T) is due to Levinson
[2], who showed that
N0(T) ^ ociV(T) A0.28.1)
for large enough T, with a = 0342. The underlying idea is to relate the
distribution of zeros of ((s) to that of the zeros of C'(s). To put matters in
f Potter and Titchmarsh A).

10.28 ZEROS ON THE CRITICAL LINE 287
their proper perspective we first note that Berndt [1 ] has shown that
and that Speiser A) has proved that the Riemann Hypothesis is
equivalent to the non-vanishing of ('(s) for 0 < a < \. This latter result
is related to the unconditional estimate
t:-l<o <\, Tx <t^ T2,C(s) =
= a + it:0 <a <%,Tl <t^ T2, C(s) =
+ O(log T2), A0.28.2)
zeros being counted according to multiplicity. This is due to Levinson
and Montgomery [1], who also gave a number of other interesting
results on the distribution of the zeros of C'(s).
We sketch the proof of A0.28.2). We shall make frequent reference to
the logarithmic derivative of the functional equation B.6.4), which we
write in the form
= -F(s), A0.28.3)
say. We note that F(^ + it) is always real, and that
F(s) = log(t/2n) + 0A/0 A0.28.4)
uniformly for t ^ 1 and |oi < 2. To prove A0.28.2) it suffices to consider
the case in which the numbers Tj are chosen so that ?(s) and ('(s) do not
vanish for t = Tjy —l^c^^. We examine the change in argument in
C'(s)/C(s) around the rectangle with vertices \ — 3 + iTlf \ — 3 + iT2,
— 1 + iT2, and — 1 + iTv where 3 is a small positive number. Along
the horizontal sides we apply the ideas of §9.4 to ((s) and ('(s)
separately. We note that ?(s) and ?(s) are each O(tA) for — 3 < a < 1.
Moreover we also have |C(— 1 + iT})\ > Tj, by the functional equation,
and hence also
>T*\ogTj,
by A0.28.3) and A0.28.4). The method of § 9.4 therefore shows that arg ?(s)
and arg ?'(s) both vary by O(log T^ on the horizontal sides of the

288 ZEROS ON THE CRITICAL LINE Chap. X
rectangle. On the vertical side a = — 1 we have
by A0.28.3) and A0.28.4), so that the contribution to the total change in
argument is 0A). For the vertical side a = \ — b we first observe from
A0.28.3) and A0.28.4) that
A0.28.5)
if t ^ T1 with T1 sufficiently large. It follows that
A0.28.6)
for T1 < t < T2, if 3 — 3{T2) is small enough. To see this, it suffices to
examine a neighbourhood of a zero p = \ + iy of C(s). Then
C(s) s - p
where m ^ 1 is the multiplicity of p. The choice s = \ + it with t -> y
therefore yields R(m') ^ 1, by A0.28.5). Hence, on taking s = \ - 3 + it,
we find that
for \s — p\ small enough. The inequality A0.28.6) now follows. We
therefore see that arg ?'(S)/C(s) varies by 0A) on the vertical side
R(s) = \ — 3 of our rectangle, which completes the proof of A0.28.2).
If we write N for the quantity on the left of A0.28.2) it follows that
NoiTJ-N^TJ = {N(T2)-N(T1)}-2N + O(log T2), A0.28.7)
so that we now require an upper bound for N. This is achieved by
applying the 'mollifier method' of §§9.20-24 to CO- -«)• Let v(<r, Tv T2)
denote the number of zeros of CO — s) in the rectangle a < R(s) < 2,
T1 < I(s) < T2. The method produces an upper bound for
2
f v(<r, TlfT2)da, A0.28.8)
u
which in turn yields an estimate N < c {N(T2) - N(TX)} for large T2. The
constant c in this latter bound has to be calculated explicitly, and must

10.28 ZEROS ON THE CRITICAL LINE 289
be less than \ for A0.28.7) to be of use. This is in contrast to (9.20.5), in
which the implied constant was not calculated explicitly, and would
have been relatively large. It is difficult to have much feel in advance for
how large the constant c produced by the method will be. The following
very loose argument gives one some hope that c will turn out to be
reasonably small, and so it transpires in practice.
In using A0.28.8) to obtain a bound for N we shall take
u = $-a/log T2,
where a is a positive constant to be chosen later. The zeros p' = /?' + iy' of
CO-— s) have an asymmetrical distribution about the critical line.
Indeed Levinson and Montgomery [1] showed that
whence /?' is \ — (loglog y')/log y' on average. Thus one might reasonably
hope that a fair proportion of such zeros have /?' < u, thereby making the
integral A0.28.8) rather small.
We now look in more detail at the method. In the first place, it is
convenient to replace ('A — s) by
«.) +
say. If we write h(s) = n'^8 r(?s) then A0.28.3), together with the
functional equation B.6.4), yields
F(s)h(s)G(s)
CA"S)= h(l-s) '
so that G(s) and ('A ~~ s) have the same zeros for t large enough. Now let
Hs)= Z bnn-° A0.28.9)
n < y
be a suitable 'mollifier' for G(s), and apply Littlewood's formula (9.9.1) to
the function G(s)\j/(s) and the rectangle with vertices u + iTl, 2 + iT2,
2 + iTv u + iT2. Then, as in §9.16, we find that
2
J v(<7, Tv T2)da
<^^[ log \G(u + it)ij/(u + it)\dt + O(logT2).
Zna J
r,

290 ZEROS ON THE CRITICAL LINE Chap. X
Moreover, as in §9.16 we have
log | G(u + it) \j/(u + it) \dt
T
1 2
Hence, if we can show that
u + it)if/(u + it)\2 dt ~ c(a)(T2~T1) A0.28.10)
for suitable Tv T2, we will have
A0.28.11)
\ 2a
whence
N (T )-N (T ) > (l ~b "v~y I om \IN(T W
by A0.28.7).
The computation of the mean value A0.28.10) is the most awkward
part of Levinson's argument. In [2] he takes y = T2*~? and
This leads eventually to A0.28.10) with
9/l 1\1 125 7a
da) = e2a 1 1
V } V2°3 24a / 2°3 °2 24a I2 12"
The optimal choice of a is roughly a = 1-3, which produces A0.28.1) with
= 0-342.
The method has been improved slightly by Levinson [4], [5], Lou [1]
and Conrey [1 ] and the best constant thus far is a = 0-3658 (Conrey [1 ]).
The principal restriction on the method is that on the size of y in
A0.28.9). The above authors all take y = T2*~?, but there is some scope
for improvement via the ideas used in the mean-value theorems G.24.5),
G.24.6), and G.24.7).

10.29 ZEROS ON THE CRITICAL LINE 291
10.29. An examination of the argument just given reveals that the
right hand side of A0.28.11) gives an upper bound for N + N*, where
N* = # {s = i + it: T,<t^ T2, C(s) = 0},
(zeros being counted according to multiplicities). However it is clear
from A0.28.3) and A0.28.4) that ?'(? + it) can only vanish if C(? + it) does.
Consequently, if we write iV(r) for the number of zeros of ((s) of
multiplicity r, on the line segment s = ^ + it, Tx < t < T2, we will have
N* = ? (r-l
r=2
Thus A0.28.7) may be replaced by
r=3
If we now define N^iT) in analogy to iV(r), but counting zeros \ + it with
0 < t < T, we may deduce that
A0.29.1)
r=3
for large enoughT, and a = 0342. In particular at least a third of the non-
trivial zeros of ?(s) not only lie on the critical line, but are simple. This
observation is due independently to Heath-Brown [5] and Selberg
(unpublished). The improved constants a mentioned above do not all
allow this refinement. However it has been shown by Anderson [1 ] that
A0.29.1) holds with a = 03532.
10.30. Levinson's method can be applied equally to the derivatives
^m)(s) of the function ?(s) given by B.1.12). One can show that the zeros
of these functions lie in the critical strip, and that the number of them,
Nm(T) say, for 0 < t < T, is N(T) + Om(log T). If the Riemann hypo-
hypothesis holds then all these zeros must lie on the critical line. Thus it is of
some interest to give unconditional estimates for
lim inf NJT)-1 # {t: 0 < t < T, ?"*>(? + it) = 0} = am,
say. Levinson [3], [5] showed that ax ^ 0-71, and Conrey [1] improved
and extended the method to give <xl ^ 0-8137, a2 ^ 0-9584 and in general
«m =

XI
THE GENERAL DISTRIBUTION OF
THE VALUES OF {(*)
11.1. In the previous chapters we have been concerned almost entirely
with the modulus of ?(s), and the various values, particularly zero,
which it takes. We now consider the problem of ?(s) itself, and the
values of s for which it takes any given value a.f
One method of dealing with this problem is to connect it with the
famous theorem of Picard on functions Avhich do not take certain values.
We use the following theorem :J
If f(s) is regular and never 0 or 1 in \s—so\ < r, and \f(so)\ ^ a,
then |/E)| < A(cc,d)for \s—so\ < dr, where 0 < 0 < 1.
From this we deduce
Theorem 11.1. ?(s) takes every value, with one possible exception, an
infinity of times in any strip 1—S < cr < 1-f S-
Suppose, on the contrary, that ?(<s) takes the distinct values a and 6
only a finite number of times in the strip, and so never above t = t0, say.
Let T > tQ-\-\, and consider the function f(s) = {?,(s)—a}/(b—a) in the
circles C, C, of radii %8 and ^S @ < S < 1), and common centre
so= 1+iS+iT. Then
and f(s) is never 0 or 1 in C. Hence
in C, and so |?(a+*T)| < A(a,b,u) for 1 < a < 1 + p, T > ^+1.
Hence ?,(s) is bounded for a > 1, which is false, by Theorem 8.4 (A).
This proves the theorem.
We should, of course, expect the exceptional value to be 0.
If we assume the Riemann hypothesis, we can use a similar method
inside the critical strip; but more detailed results independent of the
Riemann hypothesis can be obtained by the method of Diophantine
approximation. We devote the rest of the chapter to developments of
this method.
t See Bohr A)-A4), Bohr and Courant A), Bohr and Jessen A), B), E), Bohr and
Landau C), Borchsenius and Jessen A), Jessen A), van Kampen (l),van Kampen and
Wintrier A), Kershner A), Kershner and Wintner A), B), Wintner (l)-D).
% See Landau's Ergebnisse der Funktionentheorie, § 24, or Valiron's Integral Functions,
Ch. VI. § 3.

11.2 DISTRIBUTION OF THE VALUES OF ?(«) 293
11.2. We restrict ourselves in the first place to the half-plane a > 1;
and we consider, not ?(s) itself, but log ?(s), viz. the function defined
for a > 1 by the series
log?(«)= -
v
We consider at the same time the function
We observe that both functions are represented by Dirichlet series,
absolutely convergent for or > 1, and capable of being written in the
where fn(z) is a power-series in z whose coefficients do not depend on s.
In fact L{z) = _logA_2)> fn{z) = ziogpJ{1_z)
in the above two cases. In what follows F(s) denotes either of the two
functions.
11.3. We consider first the values which F(s) takes on the line or = or0,
where a0 is an arbitrary number greater than 1. On this line
and, as t varies, the arguments — t\ogpn are, of course, all related.
But we shall see that there is an intimate connexion between the set U
of values assumed by F(s) on a = a0 and the set V of values assumed
by the function
Q(o0,0lt02,...)= | fn{Pna°e™en)
n=l
of an infinite number of independent real variables 6X, 62,
We shall in fact show that the set U, which is obviously contained in V,
is everywhere dense in V, i.e. that corresponding to every value v in V
(i.e. to every given set of values 6V 62,...) and every positive e, there exists
a t such that , -p, , -., ,
\J! {cro-\-it) — v\ < e.
Since the Dirichlet series from which we start is absolutely convergent
for a = or0, it is obvious that we can find N = N(a0, e) such that
for any values of the /xn, and in particular for /xn = 6n, or for

294 THE GENERAL DISTRIBUTION OF Chap. XI
Now since the numbers logpn are linearly independent, we can, by
Kronecker's theorem, find a number t and integers gv g2,..., gN such that
\-^Ogpn-27rdn-2rrgn\ < r) (n = 1, 2,..., N),
rj being an assigned positive number. Since fn(Pna°e?nid) is, for each n,
a continuous function of 6, we can suppose 77 so small that
I {L{a^^)f{^}\ ? A1.3.2)
I
The result now follows from A1.3.1) and A1.3.2).
11.4. We next consider the set W of values which F{s) takes 'in the
immediate neighbourhood' of the line a = a0, i.e. the set of all values
of w such that the equation F(s) = w has, for every positive S, a root
in the strip \a—ao\ < S.
In the first place, it is evident that U is contained in W. Further,
it is easy to see that U is everywhere dense in W. For, for sufficiently
small S (e.g. for S < \{aQ— 1)),
\F'(s)\<K(a0)
for all values of s in the strip \a—ao\ < S, so that
li^o+^-i^+^l <#(<70)|a1-<70| (ta-orol <S). A1.4.1)
Now each value w in W is assumed by F(s) either on the line a = a0,
in which case it is a u, or at points a^it arbitrarily near the line, in
which case, in virtue of A1.4.1), we can find a u such that
—cro| < €.
We now proceed to prove that W is identical with V. Since U is con-
contained in and is everywhere dense in both V and W, it follows that
each of V and W is everywhere dense in the other. It is therefore
obvious that W is contained in F, if V is closed.
We shall see presently that much more than this is true, viz. that V
consists of all points of an area, including the boundary. The following
direct proof that V is closed is, however, very instructive.
Let v* be a limit-point of V, and let vv (v = 1, 2,...) be a sequence of
v's tending to v*. To each vv corresponds a point Pv@ljV, d2v,...) in the
space of an infinite number of dimensions defined by 0 < 6nv < 1
(n = 1, 2,...), such that O(cr0, dlv,...) = vv.
Now since (Pv) is a bounded set of points (i.e. all the coordinates are
bounded), it has a limit-point P* @*, 6%,...), i.e. a point such that from
(Pv) we can choose a sequence (PVr) such that each coordinate 6nVr of Pv,
tends to the limit 6* as r ->> 00.

11.4 THE VALUES OF ?(«) 295
It is now easy to prove that P* corresponds to v*, i.e. that
O(a0, 0*,...) = v*,
so that v* is a point of V. For the series for vVr, viz.
l
is uniformly convergent with respect to r, since (by Weierstrass's
If-test) it is uniformly convergent with respect to all the 0's; further,
the nth term tends to fn{pna°e2nie") as r -» oo. Hence
00
r—>oo j—»-oo n -— 1
which proves our result.
To establish the identity of V and W it remains to prove that V is
contained in W. It is obviously sufficient (and also necessary) for this
that W should be closed. But that W is closed does not follow, as might
perhaps be supposed, from the mere fact that W is the set of values
taken by a bounded analytic function in the immediate neighbourhood
of a line. Thus e~z* is bounded and arbitrarily near to 0 in every strip
including the real axis, but never actually assumes the value 0. The
fact that W is closed (which we shall not prove directly) depends on
the special nature of the function F(s).
Let v = O(cr0, 01} 62,...) be an arbitrary value contained in V. We
have to show that v is a member of W, i.e. that, in every strip
F(s) assumes the value v.
Let G{s) = tjniP?***)*
so that G(a0) — v. We choose a small circle C with centre a0 and radius
less than 5 such that G(s) =f= v on the circumference. Let m be the
minimum of \G(s) — v\ on C.
Kronecker's theorem enables us to choose t0 such that, for every s
inC' \F(s+ito)-G(s)\ <m.
The proof is almost exactly the same as that used to show that U is
everywhere dense in V. The series for F(s) and 6r(s) are uniformly
N
convergent in the strip, and, for each fixed N, 2/n(^CTe27rt/i") is a
i
continuous function of a, ft1}..., /%. It is therefore sufficient to show
that we can choose t0 so that the difference between the arguments
of p~s at s = cro-\-ito and p~se2iTien at s = a0, and consequently that

296 THE GENERAL DISTRIBUTION OF Chap. XI
between the respective arguments at every pair of corresponding points
of the two circles is (mod27r) arbitrarily small for n = 1, 2,..., N. The
possibility of this choice follows at once from Kronecker's theorem.
We now have
F(s+ito)-v = {G(s)-v}+{F(s+ito)-G(s)},
and on the circumference of C
\F{s+itQ)-G{s)\ <m^ \G(s)-v\.
Hence, by Rouche's theorem, F(s + it0) — v has in C the same number of
zeros as G(s) — v, and so at least one. This proves the theorem.
11.5. We now proceed to the study of the set V. Let Vn be the set
of values taken by fn{Pns) for a = a0, i.e. the set taken by fn(z) for
\z\ = Pna°- Then V is the 'sum' of the sets of points Vv V2,..., i.e. it is
the set of all values v1Jrv2Jr..., where vx is any point of Vx, v2 any point
of V2, and so on. For the function log ?(s), Vn consists of the points of
the curve described by —log(l—z) as z describes the circle \z\ = Pna°'>
for ?'(s)/?(s) it consists of the points of the curve described by
We begin by considering the function ?'(s)/?(s). In this case we can
find the set V explicitly. Let
wn — .
As zn describes the circle \zn\ = Pna°, wn describes the circle with centre
= Pn g Pn
1p'^o
and radius
Let wn = cn+wn = cn-\-Pnei^,
and let c=2cn = |i^
Then V is the set of all the values of
f
f pneiK
n=l
for independent (f>1} <f>2,.... The set V of the values of
'sum' of an infinite number of circles with centre at the origin, whose
radii plt p2,-~ form, as it is easy to see, a decreasing sequence. Let
V'n denote the nth. circle.

11.5 THE VALUES OF ?(«) 297
Then F^+^2 is the area swept out by the circle of radius p2 as its
centre describes the circle with centre the origin and radius px. Hence,
since p2 < p1, V\ 4- V2 is the annulus with radii p1— p2 and px 4- p2.
The argument clearly extends to any finite number of terms. Thus
^v consists of all points of the annulus
N N
Pi- 2 Pn < M < 2 pn,
n=2 n=l
or, if the left-hand side is negative, of the circle
M
It is now easy to see that
n =
(i) if px > p2+P3+---> the set V consists of all points w of the annulus
Pi—
00
1 Pn$
n = 2
: M ,
00
71 =
. Pn\
1
(ii) if px ^ P2JrP3Jr---j V consists of all points w for which
M < 2 Pn.
l
n =
For example, in case (ii), let w0 be an interior point of the circle. Then
we can choose N so large that
oo JY
2 Pn < 2 Pn—Kl-
N + l n = l
00
Hence w1 = w0— 2
lies within the circle V'1-\-...-\-V'N for any values of the <j>n, e.g. for
</>iY+1 = ... = 0. Hence A-
^1= 2 Pn^n
l
n
for some values of </>l5..., <j>n, and so
00
•3/' ^" n pi'Pn
w0 — 2, Pne r
as required. That V also includes the boundary in each case is clear
on taking all the <f>n equal.
The complete result is that there is an absolute constant D = 2-57...,
determined as the root of the equation
>-.Dlr»rr9 ^^-.DlOg^
>-2Z) »
1 —2~27) 2* 1—

298 THE GENERAL DISTRIBUTION OF Chap. XI
such that for cr0 > D we are in case (i), and for 1 < a0 < D we are in
case (ii). The radius of the outer boundary of V is
= ?'Ba) ?>)
in each case; the radius of the inner boundary in case (i) is
r = 2Pl—R = 21-or« log 2/A — 2-2or»)—i2.
Summing up, we have the following results for ?'(<s)/?(,s).
Theorem 11.5 (A). Tfo values which ?'(<s)/?(,s) takes on the line
a = a0 > 1 form a set everywhere dense in a region R{a0). If a0 > D,
R{cr0) is the annulus (boundary included) with centre c and radii r and R;
if a0 ^ D, R{cr0) is the circular area (boundary included) with centre c and
radius R; c, r, and R are continuous functions of a0 defined by
c = ?'Bao)/?Bao), R = c-?>0)/?(a0), r = 2*-". log 2/A-2-**)-*.
Further, as cr0 -» oo,
lime = limr = MmR = 0, limc/R = lim(R—r)/R = 0;
as cro->2), limr = 0; and as a0 -» 1, limR = oo, lime = ?'B)/?B).
Theorem 11.5 (B). The set of values which ?,'(s)/?{s) takes in the
immediate neighbourhood of a = a0 is identical with R{cr0). In particular,
since c tends to a finite limit and R to infinity as <r0 -» 1, ?'(s)/?,(s) takes
all values infinitely often in the strip 1 < a < l-\-S, for an arbitrary
positive 8.
The above results evidently enable us to study the set of points at
which ^'(s)l^(s) takes the assigned value a. We confine ourselves to
giving the result for a = 0; this is the most interesting case, since the
zeros of ?,'{s)/?(s) are identical with those of ?'(<s).
Theorem 11.5 (C). There is an absolute constant E, between 2 and 3,
such tJiat ?'(<s) ^ 0 for a > E, while ?'(s) has an infinity of zeros in every
strip between a = 1 and a = E.
In fact it is easily verified that the annulus R{a0) includes the origin
if a0 = 2, but not if cr0 = 3.
11.6. We proceed now to the study of log ?(s). In this case the set V
consists of the 'sum' of the curves Vn described by the points
Wn = —l
as zn describes the circle \zn\
In the first place, Vn is a convex curve. For if
u+iv = w= f{z) = f{x+iy),

11.6 THE VALUES OF ?(«) 299
and z describes the circle \z\ = r, then
Hence arctan— = arg{z/'(z)}—\n.
A sufficient condition that w should describe a convex curve as z
describes \z\ = r is that the tangent to the path of w should rotate
steadily through 2tt as z describes the circle, i.e. that arg{z/'(z)} should
increase steadily through 2tt. This condition is satisfied in the case
/(z) = —log(l—z); for zf'(z) = z/(l—z) describes a circle enclosing the
origin as z describes \z\ = r < 1.
If z = reie, and w = —log(l—z), then
rsind
u = — ilog(l — 2rcos0+r2), v = arctan .
1—rcosd
The second equation leads to
rcosd = ain2vzizcosv(r2—sin2v)i.
Hence, for real r and 6, \v\ ^ arcsinr. If cos^ and cos#2 are the two
values of cos 6 corresponding to a given v,
{l-2rcos61+r2){l-2rcos62+r2) = A-r2J.
Hence if ux and u2 are the corresponding values of u,
The curve Vn is therefore convex and symmetrical about the lines
u = — ^log(l — r2) and v = 0.
Its diameters in the u and v directions are ^Iog{(l4-r)/(l — r)} and
arcsin r.
Let cn = —^log(l—Pn2a°)
and wn = cn+w'n,
00
= 2 cn = ilog?Bcr0).
n =
Then the points w'n describe symmetrical convex figures with centre the
origin. Let V be the 'sum' of these figures.
It is now easy, by analogy with the previous case, to imagine the
result. The set V, which is plainly symmetrical about both axes, is either
(i) the region bounded by two convex curves, one of which is entirely interior
to the other, or (ii) the region bounded by a single convex curve. In each
case the boundary is included as part of the region.
This follows from a general theorem of Bohr on the 'summation' of
a series of convex curves.

300 THE GENERAL DISTRIBUTION OF Chap. XI
For our present purpose the following weaker but more obvious
results will be sufficient. The set V is included in the circle with centre
the origin and radius
00
If cr0 is sufficiently large, V lies entirely outside the circle of radius
n^ = arcsin 2-ffo + \ log 1^~
arcsin 2~a° — V ? log n_^ = arcsin 2-ffo + \ log 1^~ ^° — R.
It > arcsin pn a° >
n = 2 l_2^o
and so if cr0 is sufficiently near to 1, V includes all points inside the circle
of radius «,
arcsin ?>~°°.
n =
In particular V includes any given area, however large, if a0 is suffi-
sufficiently near to 1.
We cannot, as in the case of circles, determine in all circumstances
whether we are in case (i) or case (ii). It is not obvious, for example,
whether there exists an absolute constant D' such that we are in case
(i) or (ii) according as cr0 > 2)' or 1 < cr0 < D''. The discussion of this
point demands a closer investigation of the geometry of the special
curves with which we are dealing, and the question would appear to be
one of considerable intricacy.
The relations between U, V, and W now give us the following
analogues for log ?(s) of the results for ?'(<s)/?(,s).
Theorem 11.6 (A). On each line a = cr0 > 1 the values of log ?(s) are
everywhere dense in a region R{cr0) which is either (i) the ring-shaped area
bounded by two convex curves, or (ii) the area bounded by one convex curve.
For sufficiently large values of a0 we are in case (i), and for values of a0
sufficiently near to 1 we are in case (ii).
Theorem 11.6 (B). The set of values which log ?(s) takes in the
immediate neighbourhood of a = cr0 is identical with R{o0). In particular,
since R{cr0) includes any given finite area when a0 is sufficiently near 1,
log ?(s) takes every value an infinity of times in 1 < a < 1-f-S.
As a consequence of the last result, we have
Theorem 11.6 (C). the function ?(s) takes every value except 0 an
infinity of times in the strip 1 < a < l+§-
This is a more precise form of Theorem 11.1.

11.7 THE VALUES OF ?(«) 301
11.7. We have seen above that log ?(s) takes any assigned value a
an infinity of times in a > 1. It is natural to raise the question how
often the value a is taken, i.e. the question of the behaviour for large T
of the number Ma{T) of roots of log ?(s) = aincr>l,0<?<T. This
question is evidently closely related to the question as to how often, as
?->oo, the point {axt,a2t,...,aNt) of Kronecker's theorem, which, in
virtue of the theorem, comes (mod 1) arbitrarily near every point in the
N-dimensional unit cube, comes within a given distance of an assigned
point (bv b2,...,bN). The answer to this last question is given by the
following theorem, which asserts that, roughly speaking, the point
{a1t,...,aNt) comes near every point of the unit cube equally often, i.e.
it does not give a preference to any particular region of the unit cube.
Let a1?..., aN be linearly independent, and let y be a region of the N-
dimensional unit cube with volume Y {in the Jordan sense). Let Iy(T) be
the sum of the intervals between t = 0 and t = T for which the point P
(ax t,..., aN t) is (mod 1) inside y. Then
lim UT)jT = T.
The region y is said to have the volume T in the Jordan sense, if, given
e, we can find two sets of cubes with sides parallel to the axes, of volumes
Fx and F2, included in and including y respectively, such that
r±—e ^ r ^ r2+€.
If we call a point with coordinates of the form (a1t,...,aNt), modi,
an 'accessible' point, Kronecker's theorem states that the accessible
points are everywhere dense in the unit cube C. If now yly y2 are two
equal cubes with sides parallel to the axes, and with centres at accessible
points Px and P2, corresponding to tx and t2, it is easily seen that
lim Iyi(T) I Iy2(T)= 1.
For (a1t,...,aNt) will lie inside y2 when and only when {a-^t+t^—t-^,...}
lies inside yv
Consider now a set of p non-overlapping cubes c, inside C, of side e,
each of which has its centre at an accessible point, and q of which lie
inside y; and a set of P overlapping cubes c', also centred on accessible
points, whose union includes C and such that y is included in a union of
Q of them. Since the accessible points are everywhere dense, it is
possible to choose the cubes such that q/P and Q/p are arbitrarily near
to F. Now, denoting by X/^T1) the sum of ^-intervals in @, T)
y
corresponding to the cubes c which lie in y, and so on,
un

302 THE GENERAL DISTRIBUTION OF Chap. XI
Making T -> oo we obtain
and the result follows.
11.8. We can now prove
Theorem 11.8 (A). If a = a0 > 1 is a line on which log ?(s) comes
arbitrarily near to a given number a, then in every strip cr0—8 < o- < o-0-|-8
the value a is taken more than K(a, a0,8)T times, for large T, in 0 < t < T.
To prove this we have to reconsider the argument of the previous
sections, used to establish the existence of a root of log ?(s) = a in the
strip, and use Kronecker's theorem in its generalized form. We saw that
a sufficient condition that log ?(s) = a may have a root inside a circle
with centre <ro + ito and radius 23 is that, for a certain N and
corresponding numbers 91,..., 0N, and a certain rj = rj(a0, S, 91,..., 9N)
\-t0logpn-27Tdn-27Tgn\ <7) {n=l, 2,..., N).
From the generalized Kronecker's theorem it follows that the sum of
the intervals between 0 and T in which t0 satisfies this condition is
asymptotically equal to {7]J2tt)nT, and it is therefore greater than
\{7]J2tt)nT for large T. Hence we can select more than ^Gy/27r)iVT/8
numbers t'o in them, no two of which differ by less than 48. If now we
describe circles with the points ao-\-it'o as centres and radius 28, these
circles will not overlap, and each of them will contain a zero of log ?(s)—a.
This gives the desired result.
We can also prove
Theorem 11.8 (B). There are positive constants Kx(a) and K2(a) such
that the number Ma{T) of zeros of log ?(s)—a in a > 1 satisfies the
inequalities Kx{a)T < Ma{T) < K2{a)T.
The lower bound follows at once from the above theorem. The upper
bound follows from the more general result that ifb is any given constant,
the number of zeros of ?{s)—b in a > |+8 (8 > 0), 0 < t < T, is 0{T)
as T -> oo.
The proof of this is substantially the same as that of Theorem 9.15 (A),
the function ?(<s)—b playing the same part as ?(s) did tljere. Finally the
number of zeros of log ?(s)—a is not greater than the number of zeros
of ?(*)—ea, and so is 0{T).
11.9. We now turn to the more difficult question of the behaviour
of ?(s) in the critical strip. The difficulty, of course, is that ?(s) is no

11.9 THE VALUES OF ?(«) 303
longer represented by an absolutely convergent Dirichlet series. But by
a device like that used in the proof of Theorem 9.17, we are able to
obtain in the critical strip results analogous to those already obtained
in the region of absolute convergence.
As before we consider log ?(s). For a < 1, log ?(s) is defined, on each
line t = constant which does not pass through a singularity, by con-
continuation along this line from a > 1.
We require the following lemma.
Lemma. Iff{z) is regular for \z—zo\ < E, and
\f(z)\*dxdy = H,
then \f{z){<^h^ (\z-zo\ ^ E'< B).
For if \z'—zo\ < E',
= as J HSrd*-h]
J
\z-z'\ = r
Hence r-r r-r-
')|2 J
0 0 0
and the result follows.
Theorem 11.9. Let a0 be a fixed number in the range \ < a < 1. Then
the values which log ?(s) takes on a = cr0, t > 0, are everywhere dense in
the whole plane.
N
Let ?jv(<s) = ?(<s) XT {1—Pn3)-
71=1
This function is similar to the function ?(s)Mx(s) of Chapter IX, but
it happens to be more convenient here.
Let 8 be a positive number less than K°o~i)- Then it is easily seen
as in § 9.19 that for N ^ jV0(<t0, e), T > To = T0(N),
T
J \Z,N{<y+it) — \\2 dt < *T
i
uniformly for ao—B < a < oi+8 {ox > 1). Hence
T CTi + 8
f f \t,N{°+it)-\\% dodt < (cr1-cr0+28)€T.
1 CTo —O
Hence f f \t,N{o+it)—l\2 dadt < (cr1-a0+28)Ve
v—t CTo—o

304 THE GENERAL DISTRIBUTION OF Chap. XI
for more than (l-^/e)T integer values of v. Since this rectangle
contains the circle with centre s = a + it, where a0^ a ^ c1,
v -^4-<5 < * ^ v + ^-3, and radius 3, it is easily seen from the lemma
that we can choose 3 and e so that given 0 < r\ < 1, 0 < rj' < 1, we have
\ZN{a+it)-l\ < 77 (cr0 < a < ax) A1.9.1)
for a set of values of t of measure greater than A —t/)T, and for
N > N0(o, -n, 7)'), T > T0(N).
Let BN(s) = - | Log(l-;p-s) (a > 1),
N + l
where Log denotes the principal value of the logarithm. Then
?N(s) = expfi^s)}.
We want to show that RN(s) = Log?#(«), i.e. that \IRN(s)\ < \n, for
a > cr0 and the values of t for which A1.9.1) holds. This is true for
a = ox if ax is sufficiently large, since I-R^s)! -» 0 as ax -» oo. Also, by
A1.9.1), R^jvl5) > 0 f°r °o ^ CT ^ ^u s0 *na* I-^ivl5) niust remain
between — \n and ^?r for all values of o in this interval. This gives the
desired result.
We have therefore
for <t0 < a < (Tj, iV > iV0(o-0, ->y, ->;'), T > ^o(-^)' m a set of values of t
of measure greater than A — -q')T.
Now consider the function
FN(a0+it) = - f log(l-p-*-«),
n = l
and in conjunction with it the function of N independent variables
**@i,...>0tf) = - 2 log(l-p»'-e«^-).
n = l
Since 2 ^rT0"* is divergent, it is easily seen from our previous discussion
of the values taken by log ?(s) that the set of values of Q>N includes any
given finite region of the complex plane if N is large enough. In
particular, if a is any given number, we can find a number N and values
of the 0's such that <u <a a \
0)^@!,..., VN) = a.
We can then, by Kronecker's theorem, find a number t such that
\FN(crQ+it)—a\ is arbitrarily small. But this in itself is not sufficient to
prove the theorem, since this value of t does not necessarily make
) | small. An additional argument is therefore required.

11.9
Let
M,
n =
N
jtf + l
THE VALUES OF C(s) 305
n=M + l m =
Then, expressing the squared modulus of this as the product of con-
conjugates, and integrating term by term, we obtain
11 1
r r r
••• \®m,n\2 d0M+i—d0N =
0 0 0 n—m +i m-j
AT oc
2
n=ikf + l
which can be made arbitrarily small, by choice of M, for all N. It
therefore follows from the theory of Riemann integration of a con-
continuous function that, given e, we can divide up the (N—.M")-dimensional
unit cube into sub-cubes qv, each of volume A, in such a way that
v qv
Hence for M ^ MQ{<z) and any N > M, we can find cubes of total volume
greater than \ in which IOj^I < e.
We now choose our value of t as follows.
(i) Choose M so large, and give 6^,..., 9'M such values, that
OM@1,..., d'M) = a.
It then follows from considerations of continuity that, given e, we can
find an iW-dimensional cube with centre d'ly...,9'M and side d > 0
throughout which [rT) m n ^ n, ^ i
(ii) We may also suppose that M has been chosen so large that, for
any value of N, \<&M>N\ < ^e in certain (N—M)-dimensional cubes of
total volume greater than \.
(iii) Having fixed M and d, we can choose N so large that, for
T > T0(N), the inequality I-R^s)! < ?e holds in a set of values of t of
measure greater than A — \dM)T.
(iv) Let I(T) be the sum of the intervals between 0 and T for which
the point {-(tlogp,)^,..., -(t\ogpN)M
is (mod 1) inside one of the IV-dimensional cubes, of total volume greater
than \dM, determined by the above construction. Then by the extended
Kronecker's theorem, I(T) > \dMT if T is large enough. There are

306 THE GENERAL DISTRIBUTION OF Chap. XI
therefore values of t for which the point lies in one of these cubes, and
for which at the same time \RN(s) I < i<=- For such a value of t
)-a|< \FN{s)-a\ + \RN{s)\
and the result follows.
11.10. Theorem 11.10. Let \ < a < fi < 1, and let a be any complex
number. Let Maa^(T) be the number of zeros of log?(s)—a (defined as
before) in the rectangle a. < a < /?, 0 < t < T. Then there are positive
constants K-^a, a.,f$), K2(a,<x,f$) such that
Kx{a, cc,f$)T < Ma>0Lfi{T) < K2(a, ot,P)T (T > To).
We first observe that, for suitable values of the #'s, the series
I
71 = 1
is uniformly convergent in any finite region to the right of a = \. This
is true, for example, if 6n = \n for sufficiently large values of n; for then
I 2
7l>7lo 7l>7l0
which is convergent for real s > 0, and hence uniformly convergent in
any finite region to the right of the imaginary axis; and for any 0's
2 \VnSe27Tien\2 = ^Pn2a is uniformly convergent in any finite region to
the right of a = \.
If a is any given number, and the 0's have this property, we can
choose nx so large that
~ f logU-^Pn'e2"*-) <6 (or =
n=7ix + l
and at the same time so that the set of values of
1
71 = 1
includes the circle with centre the origin and radius |a| + |e|. Hence by
choosing first 9ni+1,..., and then Qly..., Qni, we can find values of the 0's,
say 6'x, 6'2,..., such that the series
= - I Iog(l-p-*e2"ie'»)
71 = 1
is uniformly convergent in any finite region to the right of a = \, and
= a.

11.10 THE VALUES OF C(s) 307
We can then choose a circle C of centre \<x-\-\fi and radius p < \{f$—a)
on which G{s) ^ a.
Let m = min \G(s)—a\.
sonC
N
Now let q)MN(s)=— j log(l— p-se27ri0»).
Then, as in the previous proof,
i i
II I<W«) I2 d6M+i- d6N dadt <A
0 o |8-$a-$j3|<|(j3)
Hence for M > ilfo(e) and any N > Jf we can find cubes of total
volume greater than \ in which
i«-i«-ij3i<
and so in which (by the lemma of § 11.9)
We also want a little more information about RN(s), viz. that RN{s)
is regular, and |-Kiv(<s) | < rj, throughout the rectangle
< i(j8-a), «o~i < t <
for a set of values of t0 of measure greater than A — r]')T. As before it
is sufficient to prove this for ?,N(s)—l, and by the lemma it is sufficient
to prove that p <o+i
<f>(to) = jdcr J \?N(8)-l\*dt<e
a to-l
for such t0, by choice of N. Now
T J3 ,
J <?(g c» = J da J rf«0 J
1 1 <
J J J
ex 1 <0-l
<Jrf(rJ |^(S)-1|2^ J dto= 2J dor J
ex 1 t-1 oc 1
by choice of N as before. Hence the measure of the set where <f>(t0) > Ve
is less than VeT, and the desired result follows.

308 THE GENERAL DISTRIBUTION OF Chap. XI
It now follows as before that there is a set of values of t0 in @, T), of
measure greater than KT, such that for \s—\<x—\fl\ < ?(j8—a)
M M
2 \og(l-p-8e2"i9'»)- 2 logil-p-3-**) < \m,
71 = 1 71=1
and also \RN{s-\-itQ)\ < \m.
At the same time we can suppose that M has been taken so large that
M
71 = 1
Then |log?(s)-G(s)| < m
on the circle with centre ^<x-\-^C-\-it0 and radius p. Hence, as before,
log?(s)—a has at least one zero in such a circle. The number of such
circles for 0 < t0 < T which do not overlap is plainly greater than KT.
The lower bound for Maoip(T) therefore follows; the upper bound holds
by the same argument as in the case a > 1.
It has been proved by Bohr and Jensen, by a more detailed study of
the situation, that there is a K(a, a, j8) such that
An immediate corollary of Theorem 11.10 is that, if Naoip(T) is the
number of points in the rectangle |<a<cr<j8<l,O<2<T where
?($) = a (a ^ 0), then
Na^(T)>K(a,a,p)T (T>T0).
For ?(s) = a if log ?(s) = log a, any one value of the right-hand side
being taken. This result, in conjunction with Theorem 9.17, shows that
the value 0 of ?(s), if it occurs at all in a > \, is at any rate quite
exceptional, zeros being infinitely rarer than a-values for any value of a
other than zero.
NOTES FOR CHAPTER 11
11.11. Theorem 11.9 has been generalized by Voronin [1], [2], who
obtained the following 'universal' property for ?(s). Let Dr be the closed
disc of radius r < \, centred at s = f, and let / (s) be any function
continuous and non-vanishing on Dr, and holomorphic on the interior
of Dr. Then for any ? > 0 there is a real number t such that
max |C(s + it) -f(s)\ < e. A1.11.1)
seD,

11.11 THE VALUES OF C(s) 309
It follows that the curve
is dense in Cn, for any fixed a in the range \ < a < 1. (In fact Voronin [1 ]
establishes this for a = 1 also.) To see this we choose a point z =
(zotz1,..., zn_ x) with z0 ± 0, and take f(s) to be a polynomial for which
/((a) _ z for o ^ m < n. We then fix an R such that 0 < R
fit
<i~ k —fl» and such that f(s) is nonvanishing on the closed disc
\s — <r\ ^ R. Thus, if r = R + \<j —1|, the disc Dr contains the circle
\s — a\ = R, and hence A1.11.1) in conjunction with Cauchy's inequality
lo V 0' ' 7~> Hid A
Rm |Z_Z()| = .
yields
. m! .
, ^-r^ /7i ' "Dm /
Hence yB) comes arbitrarily close to z. The required result then follows,
since the available z are dense in Cn.
Voronin's work has been extended by Bagchi [1 ] (see also Gonek [1 ])
so that Dr may be replaced by any compact subset D of the strip \ < R (s)
< 1, whose complement in C is connected. The condition on /is then that
it should be continuous and non-vanishing on D, and holomorphic on
the interior (if any) of D. From this it follows that if O is any continuous
function, and hx < h2 < ... < hm are real constants, then (,{s) cannot
satisfy the differential-difference equation
unless O vanishes identically. This improves earlier results of
Ostrowski [1] and Reich [1].
11.12. Levinson [6] has investigated further the distribution of the
solutions pa = fia + iya of ?(s) = a. The principal results are that
and
# {Pa- 0^ya^T, \pa-±\ >S} = Od(T) (S > 0).
Thus (c.f. § 9.15) all but an infinitesimal proportion of the zeros of(,(s) - a
lie in the strip ? — 5 < a <\ + 5, however small S may be.

310 THE GENERAL DISTRIBUTION OF Chap. XI
In reviewing this work Montgomery (Math. Reviews 53 # 10737)
quotes an unpublished result of Selberg, namely
I (Pa-b)~TTT<X°&°eT)i- A1.12.1)
This leads to a stronger version of the above principle, in which the
infinite strip is replaced by the region
log*
where <f>(t) is any positive function which tends to infinity with t. It
should be noted for comparison with A1.12.1) that the estimate
is implicit in Levinson's work. It need hardly be emphasized that despite
this result the numbers pa are far from being symmetrically distributed
about the critical line.
11.13. The problem of the distribution of values of ?(?• + it) is rather
different from that of ?(<7 + it) with ? < a < 1. In the first place it is not
known whether the values of ?(? + it) are everywhere dense, though one
would conjecture so. Secondly there is a difference in the rates of
growth with respect to t. Thus, for a fixed a > \, Bohr and Jessen A), B)
have shown that there is a continuous function F(z; a) such that
m{te[-T, T]: logt(<r +it)eR}-+ \F(x + iy; <j)dxdy (T-> oo)
2T
for any rectangle R c c whose sides are parallel to the real and
imaginary axes. Here, as usual, m denotes Lebesgue measure, and
logC(s) is defined by continuous variation along lines parallel to the
real axis, using A.1.9) for a > 1. By contrast, the corresponding result
for a = \ states that
(T-+ao).
(The right hand side gives a 2-dimensional normal distribution with
mean 0 and variance 1.) This is an unpublished theorem of Selberg,
which may be obtained via the method of Ghosh [2].

11.13 THE VALUES OF C(s) 311
By using a different technique, based on the mean-value bounds of
§7.23, Jutila [4] has obtained information on 'large deviations' of
log \(,(\ + it)\. Specifically, he showed that there is a constant A > 0 such
that
m{te[0, T):\Ui + it)\ > V}
uniformly for 1 ^ V ^ log T .

XII
DIVISOR PROBLEMS
12.1. The divisor problem of Dirichlet is that of determining the
asymptotic behaviour as x -> oo of the sum
D(x) = 2 d(n),
where d(n) denotes, as usual, the number of divisors of n. Dirichlet
proved in an elementary way that
D{x) = x\ogx+By— l)x+O(xl). A2.1.1)
In fact
= 111=
m<Vx
and A2.1.1) follows. Writing
D(x) = x\ogx-\-By— l)
we thus have A(x) = 0(x*). A2.1.2)
Later researches have improved this result, but the exact order of
A(x) is still undetermined.
The problem is closely related to that of the Riemann zeta-function.
By C.12.1) with an = d{n), s = 0, T -> oo, we have
i r tw
= —. ?{w) — dw (Ol),
ztti J w
C —loo
provided that x is not an integer. On moving the line of integration to
the left, we encounter a double pole at w = 1, the residue being
x\ogx+{2y—l)x, by B.1.16). Thus
A(x)
= --=-. t?{w) — dw @ < c' < 1).
2tti J w
' i

12.1 DIVISOR PROBLEMS 313
The more general problem of
A(*) = I dk(n),
where dk(n) is the number of ways of expressing n as a product of k
factors, was also considered by Dirichlet. We have
C+iaO
J
w
c—iao
Here there is a pole of order k a,t w = 1, and the residue is of the form
xPk(\ogx), where Pk is a polynomial of degree k—\. We write
Dk{x) = zPk(\ogx)+Ak(x), A2.1.3)
so that A2(x) = A(x).
The classical elementary theoremf of the subject is
Ak(x) = OC^-^log*-2^) (k = 2, 3,...,). A2.1.4)
We have already proved this in the case k = 2. Now suppose that it
is true in the case k— 1. We have
2 4-iN+ I 2 **-i(
n^x/m x1/*<m<x ^/
22
^1/* n<x/m
Let us denote by ^?ft(z) a polynomial in z, of degree &—1 at most, not
always the same one. Then
Hence V -P*-i(-) =
Also
f See e.g. Landau F).

314 DIVISOR PROBLEMS Chap. XII
The next term is
x
xD
N
_x y
where N = [x1-1^]. Now
and
A*-l(")
Finally
This proves A2.1.4).
We may define the order ock of Ak(x) as the least number such that
Ak{x) = O(x«*+e)
for every positive e. Thus it follows from A2.1.4) that
aft<^lT (* = 2»3--)- A2.1.5)
The exact value of ock has not been determined for any value of h.
12.2. The simplest theorem which goes beyond this elementary
result is
Theorem 12.2.f
< (* = 2, 3, 4,...).
Take an = dk(n), tfj(n) — n€, a — k, s = 0, and let x be half an odd
integer, in Lemma 3.12. Replacing w by s, this gives
c+iT
J
i
J ()
c—iT
f Voronol A), Landau E).

12.2 DIVISOR PROBLEMS 315
Now take the integral round the rectangle —a—iT, c—iT, c-\-iT,
—a+iT, where a > 0. We have, by E.1.1) and the Phragme'n-Lindelof
principle, ^
in the rectangle. Hence
c + iT
f ^{s)-d8=0[ f ^(a
-a+iT ^-a
since the integrand is a maximum at one end or the other of the range
of integration. A similar result holds for the integral over
{—a—iT, c—iT).
The residue at s = 1 is xPk(\.ogx), and the residue at s = 0 is
C*@) = 0A).
Finally
-a + iT -a + iT
J.._ S J._. S
-a-iT -a'-iT
-a + iT
f
I n1
i
s s
-a-iT
. a^dk{n) fxk(—a+it), vv/j,
x~a > -^—^ ^-^ ~ (nx)%t dt.
A, n1+a J — a+it v '
71 — 1 /7T
For 1 ^ ^< T,
The corresponding part of the integral is therefore
T
— Wk f eVd(-logt+log
1
provided that {a-\-\)k > 1. This integral is of the form considered in
Lemma 4.5, with
F{t) = kt{—log t+log2Tr+1)+tlognx.
Since F''(t) = -^-^,
the integral is

316 DIVISOR PROBLEMS Chap. XII
uniformly with respect to n and x. A similar result holds for the integral
over (—T, — 1), while the integral over (—1,1) is bounded. Hence
00
n1+a
n=l
xa J
Taking c = 1 + e, a = e, the terms are of the same order, apart from
Hence &k{x) =
The restriction that x should be half an odd integer is clearly unnecessary
to the result.
12.3. By using some of the deeper results on ?(s) we can obtain a
still better result for k ^ 4.
Theorem 12.3.f ock < ^—^ (k = 4, 5,...).
We start as in the previous theorem, but now take the rectangle as
far as a = \ only. Let us suppose that
Then ?(s) =
uniformly in the rectangle. The horizontal sides therefore give
lj d<\ =
J &{s)*ds = 0{x\) + 0\x\ f
Also J
\-%T ^ 1
Now T T
f f
f IC(i+*«)l*T< maxima)!* f
OI*
...,.dt\
Hardy and Littlewood D).

12.3 DIVISOR PROBLEMS 317
T
Also <f>(T) = J \?{i+it)\'dt =
1
by G.6.1), so that
i+iT
Hence f ?*(*)-<fo =
, J s
Altogether we obtain
Ak(x) =
The middle term is of smaller order than the last if A ^ ?. Taking
c = 1+e, the other two terms are of the same order, apart from e's, if
This gives Afc(a;) =
Taking A = ?+* (Theorems 5.5, 5.12) the result follows. Further slight
improvements for k ^ 5 are obtained by using the results stated in
§5.18.
12.4. The above method does not give any new result for k = 2 or
k = 3. For these values slight improvements on Theorem 12.2 have
been made by special methods.
27
Theorem 12.4.f a2 < —.
82
The argument of § 12.2 shows that
T
where a > 0, c > 1. Let T2j{47r2x) = IV+?, where JV is an integer, and
consider the terms with n > N. As before, the integral over 1 < t < T7
is of the form T
L r, A2.4.2)
xanl+a
J
1
van der Corput D).

318 DIVISOR PROBLEMS Chap. XII
where
= 2t{-logWlog2<jr+l)+tlognx,
F(t) =
Hence F'{t) > log-^-r, and A2.4.2) is
For n^2N this contributes to A2.4.1)
and for N < n < 2N it contributes
Similarly for the integral over — T < t < — 1; and the integral over
— 1 < t < 1 is clearly O(x~a).
If n ^ N, we write
J - J " J+/+ / + / •
— a—%T —ioo xiT —too — IT —a + iT
The first term is
loo
I f 22s7r2s-2sin2i57rr2(l-5)^!
n J s
—ioo
1 + ioo
= —2 ( cos2 \wtt Y{w)T(w~ 1 ){27T^I(nx)}2-2w dw
1— ioo
in the usual notation of Bessel functions.f
The first integral in the bracket is
t
T
which gives
f See, e.g., Titchmarsh, Fourier Integrals, G.9.8), G.9.11).

12.4 DIVISOR PROBLEMS 319
as before; and similarly for the second integral. The last two give
Altogether we have now proved that
A{x) = -~rl ^^{4W(^)}+()(|)
A2.4.3)
By the usual asymptotic formulaef for Bessel functions, this may be
replaced by
l
A2.4.4)
Now
N
= 2 V y e^i^(mnx) V
A2.4.5)
Consider the sum
We apply Theorem 5.13, with k ~ 5, and
Hence the sum is
fiY/ (m)* \A| ^(
\m\{N/m)i) l+ \\m)
Replacing N by ^iV, %N,..., and adding, the same result holds for the
sum over 1 ^ n ^ Njm. Hence the first term on the right of A2.4.5) is
Similarly the second inner sum is
and the whole sum is
f Watson, Theory of Bessel Functions, §§ 7.21, 7.23.

320 DIVISOR PROBLEMS Chap. XII
Hence, multiplying by e~ilV and taking the real part,
n =
Using this and partial summation, A2.4.4) gives
A{x) =
Taking a = e, c = 1 + e, the first and last terms are of the same order,
apart from e's, if ^y __ \x*t:)
Hence A{x) = 0{aM+e),
the result stated.
A similar argument may be applied to \{x). We obtain
, A2.4.6)
2
77 n<T31 (8tt>x) \ I
and deduce that a3 < f|.
The detailed argument is given by Atkinson C).
If the series in A2.4.4) were absolutely convergent, or if the terms
more or less cancelled each other, we should deduce that a2 ^ \; and
it may reasonably be conjectured that this is the real truth. We shall
see later that a2 ^ \, so that it would follow that a2 = \. Similarly
from A2.4.6) we should obtain a3 = \\ and so generally it may be
conjectured that ¦,_,
0Ck=='W'
12.5. The average order ofAk(x). We may define fik, the average order
of Ak(x), to be the least number such that
X
I J A2k(y) dy = O(
o
for every positive e. Since
X X
- f Afty) dy = - f O(y*«*+') dy = O(z*">+<),
x J x J
J J
0 0
we have jSA < ak for each k. In particular we obtain a set of upper
bounds for the fik from the above theorems.
As usual, the problem of average order is easier than that of order,
and we can prove more about the fik than about the <xk. We shall first
prove the following theorem.f
t Titchmarsh B2).

12.5 DIVISOR PROBLEMS 321
Theorem 12.5. Let yk be the lower bound of positive numbers a for
which m
J
— oo
A = yk; and
Sl^?WSU <i2-5-2)
— oo 0
provided that a > f3k.
c + iT
We have Dk(x) = —: lim ^~^xs ds (c > 1).
2ttI T-+ao J S
c-iT
Applying Cauchy's theorem to the rectangle y—iT, c—iT, c-\-iT,
y-\-iT, where y is less than, but sufficiently near to, 1, and allowing for
the residue at 5 = 1, we obtain
y + iT
1 f Tkl \
Ak(x) = —- lim kJUltfds. A2.5.3)
2ttI T-+oo J S
y—iT
Actually A2.5.3) holds for yk < y < 1. Forf ?k(s)js -> 0 uniformly as
t -> ioo in the strip. Hence if we integrate the integrand of A2.5.3)
round the rectangle y—iT, y—iT, y-\-iT, y -\-iT, where
Yk < V < y < l>
and make T -> oo, we obtain the same result with y instead of y.
If we replace x by l/x, A2.5.3) expresses the relation between the
Mellin transforms
J{X) — lSk{l/X), ${S) — C, \S)/S,
the relevant integrals holding also in the mean-square sense. Hence
Parse val's formula for Mellin transforms J gives
00 00 00
r
'-i dx =
h J
0 ° A2.5.4)
provided that yk < y < 1.
It follows that, if yk < y < 1,
f A^(a;)a:-2>'-1 dx < K =
i
f AJ(a;) da; < iCZ2>'+1,
t By an application of the lemma of § 11.9.
% See Titchmarsh, Theory of Fourier Integrals, Theorem 71.

322 DIVISOR PROBLEMS Chap. XII
and, replacing X by \X, \X,..., and adding,
x
A\{x) dx <
Hence fik < y, and so fik < yk.
The inverse Mellin formula is
00 00
±-^ = Akl-\z*-1dx= A^x-3-1 dx. A2.5.5)
5 J \xl J
o o
The right-hand side exists primarily in the mean-square sense, for
yk < a < 1. But actually the right-hand side is uniformly convergent
in any region interior to the strip f3k < a < 1; for
x , x x ^
X i X X
f lA^aOla;-*-1 dx < f A\{x) dx f a;-2ff-2 da;
ii lii ?x
and on putting X = 2, 4, 8,..., and adding we obtain
00
I I i-\kyU/J |^/ UiJU <^ il.
1
It follows that the right-hand side of A2.5.5) represents an analytic
function, regular for jSA < a < 1. The formula therefore holds by
analytic continuation throughout this strip. Also (by the argument
just given) the right-hand side of A2.5.4) is finite for jSA < y < 1.
Hence so is the left-hand side, and the formula holds. Hence yk < Bk,
and so, in fact, yk = fik. This proves the theorem.
12.6. Theorem 12.6 (A).f
If \ < a < 1, by Theorem 7.2
CaT < | |C(a+»0|2^< ] \Z(v+it)\* dt\ dt
T
Hence f | t>{o+it)\2k dt ^ 2*-iCJ T.
?t
t Titchmaxsh B2).

12.6 DIVISOR PROBLEMS 323
Hence, if 0 < a < |, T > 1,
00 J< rp
-co i
T
f I ?A -cx-^) |2* dt (by the functional equation)
^ Ck~1C\_
This can be made as large as we please by choice of T if a < |(&—
Hence n ^ ^Ili
and the theorem follows.
Theorem 12.6 (B).f
For cck ^
Much more precise theorems of the same type are known. Hardy
proved first that both
A{x) > Kxi, A(x) < -Kx*
hold for some arbitrarily large values of a;, and then that x* may in each
case be replaced by {x }og ^ }oglog ^
12.7. We recall that (§ 7.9) the numbers ak are defined as the lower
bounds of a such that
T
i
We shall next prove
Theorem 12.7. For each integer k ^ 2, a necessary and sufficient
condition that ».¦,
w that ^^^W' A2'7>2)
Suppose first that A2.7.2) holds. Then by the functional equation
t t •,
J \t(a+it)\2k dt = 0^-to» J |^(l-a-^)|2*^| =
l * i
t Hardy B).

324 DIVISOR PROBLEMS Chap. XII
for a < \{k— l)jk. It follows from the convexity of mean values that
T
J \?{a+it)\2k dt =
for
The index of T is less than 2 if
A/
Then
Hence A2.5.1) holds. Hence yk ^ %{k—l)jk. Hence &
and so, by Theorem 12.6 (A), A2.7.1) holds.
On the other hand, if A2.7.1) holds, it follows from A2.5.2) that
T
| \C{a+it)\^dt= 0{T2)
i
for a > \{k—\)/k. Hence by the functional equation
T
[ \?{a+it)\2k dt = 0{T^-2^+2)
i
for a < ^(k-\-l)/k. Hence, by the convexity theorem, the left-hand
side is O(T1+C) for a = ^{k+l)jk; hence, in the notation of §7.9,
<r'k < i(^+l)/^» a»d so A2.7.2) holds.
12.8. Theorem 12.8.f
^2 — h A> = i> ^4 ^ 7-
By Theorem 7.7, ak < 1 — ljk. Since
it follows that jS2 = ?, j83 = ^.
The available material is not quite sufficient to determine /J4. Theorem
12.6 (A) gives jS4 ^ f. To obtain an upper bound for it, we observe that,
by Theorem 5.5. and G.6.1),
J |?(K»0|8«& = °(^+e J I«K*'OI4*) = O(Tf+«),
f The value of & is due to Hardy C), and that of ft, to Cramer D); for ?4 see
Titchmarsh B2).

12.8 DIVISOR PROBLEMS 325
and, since a4 ^ T7g by Theorem 7.10,
t i t
J I«ft+*OI8* = O\T* J \m-it
Hence by the convexity theorem
T
J |?(<t+&)|8 dt = (^(^
i
for i3g < o- < |. It easily follows that y4 < f, i.e. jS4 < |.
NOTES FOR CHAPTER 12
12.9. For large k the best available estimates for txk are of the shape
<xk ^ 1 - Ck "*, where Cis a positive constant. The first such result is due
to Richert [2]. (See also Karatsuba [1], Ivic [3; Theorem 13.3] and Fujii
[3].) These results depend on bounds of the form F.19.2).
For the range 4 ^ k ^ 8 one has <xk ^ J -1/& (Heath-Brown [8]) while
for intermediate values of k a number of estimates are possible (see Ivic
[3; Theorem 13.2]). In particular one has a9 ^ ff-, a10 ^ |?, an ^ ^r, and
a
< 5
l2 ^ 7-
12.10. The following bounds for a2 have been obtained.
^j- = 0-330000 ... van der Corput B),
|J= 0-329268 ... van der Corput D),
if = 0-326086 ... Chih [1], Richert [1],
?f = 0-324324 ... Kolesnik [1 ],
= 0-324273 ... Kolesnik [2 ],
= 0-324074 ... Kolesnik [4],
= 0-324009 ... Kolesnik [5 ].
In general the methods used to estimate a2 and /i(^) are very closely
related. Suppose one has a bound
1 3 3 onl
J + cx-l(mnJ})<(MN)^x2 *
Af<m=sAf.
A2.10.1)
for any constant c, uniformly for M < M1 ^ 2M, iV < Nx ^ 2iV, and
MN ^ a;2- 4a. It then follows that /i(}) ^ p, a2 ^ 5, and E(T) < T 9+E (for
E(T) as in §7.20). In practice those versions of the van der Corput

326 DIVISOR PROBLEMS Chap. XII
method used to tackle /*(?) and a2 also apply to A2.10.1), which explains
the similarity between the table of estimates given above and that
presented in §5.21 for /*(?). This is just one manifestation of the close
similarity exhibited by the functions E(T) and A(x), which has its origin
in the formulae G.20.6) and A2.4.4). The classical lattice-point problem
for the circle falls within the same area of ideas. Thus, if the bound
A2.10.1) holds, along with its analogue in which the summation
condition m = 1 (mod 4) is imposed, then one has
# {(m, n)elz: mz+n2 ^ x) = nx + O(x®+t).
Jutila [3] has taken these ideas further by demonstrating a direct
connection between the size of A(x) and that of {,{\ + it) and E(T). In
particular he has shown that if a2 = ^ then n(ty ^ ^- and E(T) <? T& + E.
Further work has also been done on the problem of estimating a3. The
best result at present is a3 ^ |§-, due to Kolesnik [3]. For a4, however, no
sharpening of the bound a4 ^ \ given by Theorem 12.3 has yet been
found. This result, dating from 1922, seems very resistant to any attempt
at improvement.
12.11. The Q-results attributed to Hardy in §12.6 may be found in
Hardy [1 ]. However Hardy's argument appears to yield only
A(x) = Q + ((xlogxLloglog;c), A2.11.1)
and not the corresponding Q_ result. The reason for this is that
Dirichlet's Theorem is applicable for Q+, while Kronecker's Theorem is
needed for the Q_ result. By using a quantitative form of Kronecker's
Theorem, Corradi and Katai [1 ] showed that
»/ x ^ ( i / (loglogxL
A(x) = Q_ <X4 exp c v s s
(logloglogxL
for a certain positive constant c. This improved earlier work of Ingham
[1] and Gangadharan [1]. Hardy's result A2.11.1) has also been
sharpened by Hafner [1 ] who obtained
for a certain positive constant c. For k ^ 3 he also showed [2] that, for a
suitable positive constant c, one has
Ak(x) = Qie[(xlogxfk~1)l2k(loglogx)a exp{-c(logloglogx)

12.11 DIVISOR PROBLEMS 327
where
k-1
and Q* is Q + for k = 3 and Q ± for k ^ 4.
12.12. As mentioned in §7.22 we now have a4 ^ J, whence /?4 = J,
(Heath-Brown [8]). For & = 2 and 3 one can give asymptotic formulae for
o
Thus Tong [1] showed that
1
0
with R2(x) <^ x(logxM and
-^ . . ^ _l^ „ 3 —4G1
Taking cr3 ^ ^ (see §7.22) yields c3 ^ J^-. However the available
information concerning ak is as yet insufficient to give
ck < Bk — l)/k for any k ^ 4. It is perhaps of interest to note that Hardy's
result A2.11.1) implies R2(x) = Q{x%(logx)~%}, since any estimate
R2(x) <€ F(x) easily leads to a bound A2(x) <€ {F(x) logx}^, by an argu-
argument analogous to that given for the proof of Lemma a in § 14.13.
Ivic [3; Theorems 13.9 and 13.10] has estimated the higher moments of
A2(x) and A3(x). In particular his results imply that
o
For AJx) his argument may be modified slightly to yield
1
0
These results are readily seen to contain the estimates a2 ^ ^, /?2 ^ ? and
a3 ^ i' ?3 ^ h respectively.

XIII
THE LINDELOF HYPOTHESIS
13.1. The Lindelof hypothesis is that
?{$+it) = Op)
for every positive e; or, what comes to the same thing, that
?(<t+&) = Op)
for every positive e and every a > \; for either statement is, by the
theory of the function fx(a), equivalent to the statement that fx(a) = 0
for a ^ \. The hypothesis is suggested by various theorems in Chapters
V and VII. It is also the simplest possible hypothesis on ft(o-), for on it
the graph of y = fx(o) consists simply of the two straight lines
y = \-a (<r < 1), y = 0 (a > J).
We shall see later that the Lindelof hypothesis is true if the Riemann
hypothesis is true. The converse deduction, however, cannot be made
—in fact (Theorem 13.5) the Lindelof hypothesis is equivalent to a
much less drastic, but still unproved, hypothesis about the distribution
of the zeros.
In this chapter we investigate the consequences of the Lindelof
hypothesis. Most of our arguments are reversible, so that we obtain
necessary and sufficient conditions for the truth of the hypothesis.
13.2. Theorem 13.2.f Alternative necessary and sufficient conditions
for the truth of the Lindelof hypothesis are
T
* f \?,{$+it)\2k dt = 0{Te) (k = 1, 2,...); A3.2.1)
i
T
1 C
~m I ltvCT~r*fJI ai — v^i ) [a .> f, K — i, ^,...J, [io.z.4)
l
T
1 (a > -I, k = 1, 2,...). A3.2.3)
J n = l
The equivalence of the first two conditions follows from the convexity
theorem (§ 7.8), while that of the last two follows from the analysis of
§ 7.9. It is therefore sufficient to consider A3.2.1).
f Hardy and Littlewood E).

13.2 THE LINDELOF HYPOTHESIS 329
The necessity of the condition is obvious. To prove that it is sufficient,
suppose that ?($+*<) is not O(te). Then there is a positive number A,
and a sequence of numbers \-\-itv, such that tv->oo with v, and
1171_1_<jV ^ I —,. f^fh (C1 ~^~~ n\
On the other hand, on differentiating B.1.4) we obtain, for t > 1,
I ?'(*+**) I <Et,
E being a positive absolute constant. Hence
t
<2E\t-tv\tv <\Ctx
if \t—tv\ < tv 1 and v is sufficiently large. Hence
Take T = \tv, so that the interval {tv—t'1, t^t'1) is included in (T, 2T)
if v is sufficiently large. Then
IT tv + tv1
f r
I I niA-ii) \2k fa > I
J J
tv~t~l
which is contrary to hypothesis if k is large enough. This proves the
theorem.
We could plainly replace the right-hand side of A3.2.1) by 0{TA)
without altering the theorem or the proof.
13.3. Theorem 13.3. A necessary and sufficient condition for the truth
of the Lindelof hypothesis is that, for every positive integer k and a > \,
) = y <MHl> +o(t-x) (t > o),
z ns
ns
where 8 is any given positive number less than 1, and A = X(k, b,a) > 0.
We may express this roughly by saying that, on the Lindelof hypo-
hypothesis, the behaviour of ?(s), or of any of its positive integral powers,
is dominated, throughout the right-hand half of the critical strip, by a
section of the associated Dirichlet series whose length is less than any
positive power of t, however small. The result may be contrasted with
what we can deduce, without unproved hypothesis, from the approxi-
approximate functional equation.
Taking an — dk(n) in Lemma 3.12, we have (if a; is half an odd integer)
c + iT
ns 2iri J iv \T(a-\~c—
n<x c_iT

330 THE LINDELOF HYPOTHESIS Chap. XIII
where c > 1— a-\-€. Now let 0 < t < T—l, and integrate round the
rectangle \—a—iT, c—iT, c+iT, \—a+iT. We have
J
/r
w
J 1 — S \1 — 5
rectangle
P being a polynomial in its arguments. Also
c-iT %
(c-iT %-o+iT
f + f
^oiT c+iT
c+iT '
by the Lindelof hypothesis; and
A — <J-\-iT T
r
j
by the Lindelof hypothesis. Hence
and A3.3.1) follows on taking x = [^s]+i c = 2, T = t3.
Conversely, the condition is clearly sufficient, since it gives
*($) = O{ J n
where 8 is arbitrarily small.
The result may be used to prove the equivalence of the conditions
of the previous section, without using the general theorems quoted.
13.4. Another set of conditions may be stated in terms of the numbers
ock and fik of the previous chapter.
Theorem 13.4. Alternative necessary and sufficient conditions for the
truth of the Lindelof hypothesis are
ak^\ (k=2,S,...), A3.4.1)
fo < \ (Jc=2, 3,...), A3.4.2)
& = ^JT (*=2>3,...). A3.4.3)
As regards sufficiency, we need only consider A3.4.2), since the other

13.4 THE LINDELOF HYPOTHESIS 331
conditions are formally more stringent. Now A3.4.2) gives yk ^ \, and
so t
f \a<y+it)\2k dt = O(T*)
The truth of the Lindelof hypothesis follows from this, as in § 13.2.
Now suppose that the Lindelof hypothesis is true. We have, as in
§ 12.2, 2 + iT
] )
] ()
2-iT
Now integrate round the rectangle with vertices at \—iT, 2—iT,
2+iT, \+iT. We have
2±iT
f
-
\ + iT . T
f t*{s)-ds = O\x\ f
| -T
The residue at s = 1 accounts for the difference between Dk(x) and
Afcfc). Hence
Taking T = x2, it follows that ak < ^-. Hence also /^ < ^. But in fact
ak ^ \ on the Lindelof hypothesis, so that, by Theorem 12.7, A3.4.3)
also follows.
13.5. The Lindelof hypothesis and the zeros.
Theorem 13.5.f A necessary and sufficient condition for the truth of
the Lindelof hypothesis is that, for every a > \,
N(a, T+1)-N{o, T) = o (log T).
The necessity of the condition is easily proved. We apply Jensen's
formula 2tt
^L = JL f iog|/(^)| ^-iog 1/@I,
1...rn ltt j
o
where r1}... are the moduli of the zeros of f{s) in \s\ < r, to the circle
with centre 2-{-it and radius §—18, f{s) being t,(s). On the Lindelof
t Backhand D).

332 THE LINDELOF HYPOTHESIS Chap. XIII
hypothesis the right-hand side is less than o (log t); and, if there are
N zeros in the concentric circle of radius §—|S, the left-hand side is
greater than ^lOg{(|-i8)/(|-|8)}.
Hence the number of zeros in the circle of radius §—|S is o {logt); and
the result stated, with a = %-\-B, clearly follows by superposing a
number (depending on 8 only) of such circles.
To prove the converse,! let C± be the circle with centre 2-\-iT and
radius |—8 (8 > 0), and let X^ denote a summation over zeros of ?(s)
in Cx. Let C2 be the concentric circle of radius |—28. Then for s in CB
This follows from Theorem 9.6 (A), since for each term which is in one
of the sums , ,
V _J_ l
Z
p * s—p'
M \t-y\<l r
but not in the other, \s—p\ ^8; and the number of such terms is
Oflog T).
Let C3 be the concentric circle of radius 1—38, C the concentric circle
of radius \. Then ^r(«5) = o (log T) for s in C, since each term is 0A),
and by hypothesis the number of terms is o (log T). Hence Hadamard'e
three-circles theorem gives, for s in C3,
14,(8) |< {o (log T)}«{0(8-Mog T)Y
where a-fjS=l,0</3<l,a and /S depending on 8 only. Thus in C3
<?(*) = o (log T),
for any given 8.
Now
2
f 0(«) da =
i+38
- 21 {lo
since 2 nas ° (log ^T) terms. Also, if t = T, the left-hand side is o (log T).
Hence, putting t = T and taking real parts,
Since j^_[_3S_[_iy—^| < A in Q, it follows that
i.e. the Lindelof hypothesis is true.
f Littlewood D).

13.6 THE LINDELOF HYPOTHESIS 333
13.6. Theorem 13.6 (A).f On the Lindelof hypothesis
S{t) = o(logt).
The proof is the same as Backlund's proof (§ 9.4) that, without any
hypothesis, S(t) = O(\ogt), except that we now use ?(s) = O(te) where
we previously used ?(s) = 0{tA).
Theorem 13.6 (BL On the Lindelof hypothesis
S^t) = o (log t).
Integrating the real part of (9.6.3) from \ to ?-f 38,
? + 35 ? + 35
f log|?(«)| d<7 = I f log|«-P|*T+0(81og<),
where p = fS-\-iy runs through zeros of ?(s). Now
J log\s-P\do = ±
and
J log\a-P\da^ J lOg|a-|-|8|^a= 38(logl8-l).
Hence
f log | ?(*)| da « 2
J Iy-<|
= 0(8 log 1/8. log *).
Also, as in the proof of Theorem 13.5,
log?(«) = I^og^-pJ+oaogO (i+3S < a < 2).
Hence
j log I 5(«) I da = 21
Hence, by Theorem 9.9,
2
= 0(81ogl/8.1ogO+o(lo
and the result follows on choosing first 8 and then t.
f Cramer A), Littlewood D). % Littlewood D).

334 THE LINDELOF HYPOTHESIS Chap. XIII
NOTES FOR CHAPTER 13
13.7. Since the proof of Theorem 13.6(A) is not quite straightforward
we give the details. Let
and define n(r) to be the number of zeros of g(z) in the disc \z\ ^ r. As in
§ 9.4 one finds that S(T) <^ rc(§) +1. Moreover, by Jensen's Thorem, one
has
\" h \ -log\g@)l A3.7.1)
With our choice of g(z) we have log | g@) | = log | RCB + iT) \ = O(l). We
shall take R = § + 3. Then, on the Lindelof Hypothesis, one finds that
i!>
\((Rei
for cos 9 ^ — 3/BR) and T sufficiently large. The remaining range for 9
is an interval of length O(<5^). Here we write R(Rei& + 2) = a, so that
\ — <5 ^ a ^ \. Then, using the convexity of the \i function, together
with the facts that ^@) = \ and, on the Lindelof Hypothesis, that
= 0, we have \i(o) ^ <5. It follows that
for cos 9 ^ — 3/2R, and large enough T. We now see that the right-hand
side of A3.7.1) is at most
O(e log T) + O{dkd + e) log T).
Since
we conclude that
and on taking 3 = ?3 we obtain n(f) = O(g3 log T), from which the result
follows.
13.8. It has been observed by Ghosh and Goldston (in unpublished

13.8 THE LINDELOF HYPOTHESIS 335
work) that the converse of Theorem 13.6(B) follows from Lemma 21 of
Selberg E).
Theorem 13.8. If S^t) = o(logt), then the Linde 16f hypothesis holds.
We reproduce the arguments used by Selberg and by Ghosh and
Goldston here. Let \ ^ a ^ 2, and consider the integral
5 + ioo
2ni J 4-(s-aJ '
5 — ioo
Since log ((s + iT) <^ 2~~R(s) the integral is easily seen to vanish, by
moving the line of integration to the right. We now move the line of
integration to the left, to R(s) = a, passing a pole at s = 2 +a, with
residue — J log(B + <r + iT) = 0A). We must make detours around
s = 1 — iT, if a < 1, and around s = p — iT, if a < /?. The former, if
present, will produce an integral contributing 0{T~2), and the latter,
if present, will be
du
It follows that
log Z(a +it+ iT)dt
4 + t2
_ oo 0
for T ^ 1. We now take real parts and integrate for \ ^ a ^ 2. Then by
Theorem 9.9 we have
- oo
A3.8.1)
By our hypothesis the integral on the left is o(log T). Moreover
4-{u + i(y-T)}2) "@, otherwise.
If a>\ is given, then each zero counted by N(a, T+l) — N(<r,T)
contributes at least \{a - §JA to the sum on the right of A3.8.1), whence
N(a, T + 1)-N(a, T) = o(log T). Theorem 13.8 therefore follows from
Theorem 13.5.

XIV
CONSEQUENCES OF THE RIEMANN
HYPOTHESIS
14.1. In this chapter we assume the truth of the unproved Riemann
hypothesis, that all the complex zeros of ?(s) lie on the line a = \. It
will be seen that a perfectly coherent theory can be constructed on this
basis, which perhaps gives some support to the view that the hypothesis
is true. A proof of the hypothesis would make the 'theorems' of this
chapter essential parts of the theory, and would make unnecessary
much of the tentative analysis- of the previous chapters.
The Riemann hypothesis, of course, leaves nothing more to be said
about the 'horizontal' distribution of the zeros. From it we can also
deduce interesting consequences both about the Vertical' distribution
of the zeros and about the order problems. In most cases we obtain
much more precise results with the hypothesis than without it. But
even a proof of the Riemann hypothesis would not by any means com-
complete the theory. The finer shades in the behaviour of ?(s) would still
not be completely determined.
On the Riemann hypothesis, the function log ?(s), as well as ?(s), is
regular for a > \ (except at s = 1). This is the basis of most of the
analysis of this chapter.
We shall not repeat the words 'on the Riemann hypothesis', which
apply throughout the chapter.
14.2. Theorem 14.2.| We have
log ?(s) = 0{(log*J-2a+e} ' A4.2.1)
uniformly for % < a0 ^ a <I 1.
Apply the Borel-Caratheodory theorem to the function log ?(z) and
the circles with centre 2+it and radii |—|S and |—S @ < S < ?). On
the larger circle
R{log?B)} = log|?(«)| <A\ogt.
Hence, on the smaller circle,
<AS-nogt. A4.2.2)
f Littlewood A).

14.2 CONSEQUENCES OF RIEMANN HYPOTHESIS 337
Now apply Hadamard's three-circles theorem to the circles Cx, C2, Cz
with centre ax-\-it A < o^ ^ t), passing through the points l-\-r)-\-it,
o-\-it, \-\-h-\-it. The radii are thus
r1 = a1— 1 — rj, r2 = a1—a, rz = a1—%—S.
If the maxima of |log((,z)| on the circles are Mx, M2, M3, we obtain
where
M2
By A4.2.2), Mz < ^LS^logi; and, since
l°&Us) = 2,tf (AiN<!)' A4-2-3)
n=2 n
00
n=2
00
^ ^ max
Hence
|log^(or + *0| < (-J1 7^-^-T < ^_ ^
The result stated follows on taking S and rj small enough and ax large
enough. More precisely, we can take
a1 = - = - = loglog t;
o rj
since (log*)O(8) = e°(8lo»lo*» = eOA) = 0A),
etc., we obtain
log ?(8) = 0{loglog *(log tf-*°) A + j-^ < a < 1 j.
A4.2.4)
Since the index of log t in A4.2.1) is less than unity if e is small enough,
it follows that (with a new e)
—«log* < log|?(s)| < elog* (t > «„(«)),
i.e. we have both ^ = 0(^e)j A4.2.5)
_L = 0(?) A4.2.6)
for every a > ?. In particular, iAe <rw«A o/iAe Lindelof hypothesis follows
from that of the Riemann hypothesis.

338 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
It also follows that for every fixed a > \, as T -> oo
dt
j
I ?(*+**) I2
T.
For a > 1 this follows from G.1.2) and A.2.7). For \ < a < 1 it follows
from A4.2.6) and the analysis of § 7.9, applied to l/?(s) instead of to ?k(s).
14.3. The functionf v{o). For each a > \ we define v{o) as the
lower bound of numbers a such that
log?(«) =
It is clear from A4.2.3) that v{o) < 0 for a > 1; and from A4.2.2) that
v{a) < 1 for ? < a < 1; and in fact from A4.2.1) that v(a) < 2 —2a for
$ <<7 < 1.
On the other hand, since AxB) = 1, A4.2.3) gives
and hence v(a) > 0 if a is so large that the right-hand side is positive.
Since m
f
<
n=3 n=3
JL< fdx = 21"g
na ) xa a—I
this is certainly true for a ^ 3. Hence v(a) = 0 for a > 3.
Now let ^ < ax < a < a2 ^ 4, and suppose that
log tfa+it) = O(\ogat), log ^(a2+»0 = O(log6i).
Let <7(<s) = log ^E){log(—i,s)}~fc(s\
where k(s) is the linear function of 5 such that &(oi) = a, k(a2) — b, viz.
h(s) =
Here {log(-w)}-«s> =
where log(—w) = \og{t—io), loglog(—is) (t > e)
denote the branches which are real for a = 0. Thus
log(—is) =
loglog(-w) = {
= loglog<+0A/0-
f Bohr and Landau C), Littlewood E).

14.3 CONSEQUENCES OF RIEMANN HYPOTHESIS 339
Hence
= e-R{k(s)loeloe(--is)} = e-Ko)loglog t+O(.yt)
Hence g(s) is bounded on the lines a = a1 and a = a2; and it is 0(logKt)
for some K uniformly in the strip. Hence, by the theorem of Phragme"n
and Lindelof, it is bounded in the strip. Hence
i.e. v(a) < k(a) = V" "U"-rv "^ (U.Z.I)
Taking a = 3, a2 — 4, vC) = 0, 6 = 0, we obtain a > 0. Hence
v(cr) ^ 0 for a > \. Hence v(a) = 0 for a > 1.
Since v(a) is finite for every a > \, we can take a = ^(crjj + c,
6 = v(cr2)+€ in A4.3.1). Making c -> 0, we obtain
v{a)
i.e. v(a) is a convex function of a. Hence it is continuous, and it is non
increasing since it is ultimately zero.
We can also show that ?'(s)j?(s) has the same v-function as log ?(s)
Let v^s) be the v-function of ?'(s)/?,(s). Since
\s-z\=8
we have vx{a) <! v(o—S)
for every positive S; and since v{a) is continuous it follows that
vx{a) < v{a).
We can show, as in the case of v(a), that vx{a) is non-increasing, and
is zero for a ^ 3. Hence for a < 3
= 0( (logO
= 0{(log 0
i.e. v(a) < vx{a).

340 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
The exact value of v{a) is not known for any value of a less than 1.
All we know is
Theorem 14.3. For % < o < 1,
1-a < v(a) < 2A-a).
The upper bound follows from Theorem 14.2 and the lower bound
from Theorem 8.12. The same lower bound can, however, be obtained
in another and in some respects simpler way, though this proof, unlike
the former, depends essentially on the Riemann hypothesis. For the
proof we require some new formulae.
14.4. Theorem 14.4.| As t -*¦ oo,
), A4.4.1)
l{s) Z, ns p
uniformly for ^ ^ a <! |, e~v' <! S <! 1.
Taking an = A(n), f(s) = — ?'(s)/?(,s) in the lemma of § 7.9, we have
2 + ioo
J
2-ioo
Now, by Theorem 9.6 (A),
53 J ^s)^d, (H.4.2)
i
and there are O(logt) terms in the sum. Hence
^ = 0(log t)
on any line a =? |. Also
uniformly for — 1 ^ a ^ 2. Since each interval {n,n-\-l) contains
values of t whose distance from the ordinate of any zero exceeds
g n, there is a tn in any such interval for which
(-l<a<2, t = tn).
f Littlewood E), to the end of § 14.8.

14.4
CONSEQUENCES OF RIEMANN HYPOTHESIS
341
By the theorem of residues,
, 2 + itn i+itn i-itn 2-it
U.i + L+1+i
?{*)
~tn<y<tn
The integrals along the horizontal sides tend to zero as n -> oo, so that
~ ) If ?'(z)
Since Y(z—s) =
IOO
f e-A\y-
J
— oo
= 0
the integral is
-* dy)
J
l J + J )e-
-00 2t '
-i) = 0(S°-*\ogt).
Also
= O(e-^+iv0 = 0{e~At) =
This proves the theorem.
14.5. We can now prove more precise results about ^'(s)j^(s) and
log ?(s) than those expressed by the inequality v(a) ^ 2 — 2a.
Theorem 14.5. We have
Q& = 0{(log iJ-
uniformly for
We have
<
n=l

342 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Now
2—ioo
since we may move the line of integration to R(z) = f, and the leading
term is the residue at z = 1. Also
uniformly for a in the above range. Hence
\V{p-8)\ <A ?
y
The number of terms in the inner sum is
0{log(t+n)} = O(\ogt)+0{\og(n+l)}.
Hence we obtain
0\fe-A*{hogt+]og(n+l)}] = O(\ogt).
Hence ii^
and taking 8 = (Iog0~2 we obtain the first result.
Again for a0 ^ a ^ a1
log m = log tfa+it)- f ^?±
If a ^ a2 < ax and c < 2(a!—a2), this is of the required form; and since
a1 and so a2 may be as near to 1 as we please, the second result (with
a2 for o-j) follows.
14.6. To obtain the alternative proof of the inequality v(a) > 1—a
we require an approximate formula for log ?(s).
Theorem 14.6. For fixed a and a such that \ < a < a < 1, and
e~v< < 8 < 1,
log Us) = y
w 71

14.6 CONSEQUENCES OF RIEMANN HYPOTHESIS 343
Moving the line of integration in A4.4.2) to R(w) = a, we have
oo a + ioo
a —too
Since ?'(s)/?(,s) has the v-function v{a), the integral is of the form
§a-a J e-Aiv-%log{\y\ + 2)yw+edy =
— 00 '
and F(l— s)Bs~1 is also of this form, as in § 14.4. Hence
This result holds uniformly in the range [a, §], and so we may integrate
over this interval. We obtain
log?(s) —
n = l
00 A / \
as required.
14.7. Proof that v(a) > 1—a. Theorem 14.6 enables us to extend
the method of Diophantine approximation, already used for a > 1, to
values of a between \ and 1. It gives
00
= V
n
n =
for all values of N. Now by Dirichlet's theorem (§ 8.2) there is a number
t in the range 2tt < t ^ 2^^, and integers xly..., xN, such that
" q
Let us assume for the moment that this number t satisfies the condition
of Theorem 14.6 that e~v' < 8. It'gives
»t N i / -
/h^lcos(tlogn)e-8n > V ^
n=l n=l

344 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Now
^N i
na
\oqlN Z* n
4
n=l
1
as in § 14.5. Hence
Take q = N = [S~a], where a > 1. The second and third terms on the
right are then bounded. Also
log t < N log q+log 2tt < ^ log - + log 27r,
o o
so that S < K{\ogt)-^a+e.
Hence log|?(s)| > K(\ogtI-a-v+0{(\ogt)«-a+vM+7)'},
where rj and rj' are functions of a which tend to zero as a -> 1.
If the first term on the right is of larger order than the second, it
follows at once that v{a) > 1—a. Otherwise
a—o-\-v{oc) ^ 1—a,
and making a -> a the result again follows.
We have still to show that the t of the above argument satisfies
e~v* < S. Suppose on the contrary that S < e~v* for some arbitrarily
small values of S. Now, by (8.4.4),
\i{8)\ > (cos — -2W-°)t(cr) > -i_(i-2^1-')
\ q J a—1
for a > 1, q > 6. Taking a = 1+log 8/logiV,
|?(s)| > -^- = ^LlogiV > Alogl > At\.
Since \?{s)\ -> oo and i ^ 2tt, t -> oo, and the above result contradicts
Theorem 3.5. This completes the proof.
14.8. The function ?,(l+it). We are now in a position to obtain
fairly precise information about this function. We shall first prove
Theorem 14.8. We have
|log?(l+*OI < logloglog«+4. A4.8.1)

14.8 CONSEQUENCES OF RIEMANN HYPOTHESIS 345
In particular
it) = 0(loglogfl, A4.8.2)
A4.8.3)
Taking a = 1, a = f in Theorem 14.6, we have
n
n = \
N K .
n =
by C.14.4). Taking 8 = log-4^, N = 1 + pog5*], the result follows.
Comparing this result with Theorems 8.5 and 8.8, we see that, as far
as the order of the functions ?(l-\-it) and 1/?A-H) is concerned, the
result is final. It remains to consider the values of the constants involved
in the inequalities.
14.9. We define a function /?(ct) as
' 2-2a
By the convexity of v{o) we have, for \ < a < a < 1,
1—a 1—
i.e. p(o') < p(a).
Thus j8(ct) is non-increasing in (|, 1). We write
j8($) = lim j8
CT-^-J + O
Then by Theorem 14.3, for ^
We shall now prove f
Theorem 14.9. As t -> oo
I^(i+*OI <
1 <
t
(a), j8(l) = lim j8(a).
<a<l,
2]8(l)ey{l+o(l)}loglog*,
2j3(l) (l+o (l)}loglog^.
7T2
Littlewood F).
A4.9
A4.9
.1)
•2)

346 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
We observe that the 0A) in Theorem 14.6 is actually o(l) if 8 -> 0.
Also, taking a = 1,
81-a(
if 8 = (
Hence, for such 8,
= 2
p,m
Now the modulus of the second double sum does not exceed
Q-8pm o—Smp
z.
This is evidently uniformly convergent for 8^0, the summand being
less than p~m. Since each term tends to zero with 8 the sum is o(l).
Hence
p -Smp
p,m
The second term is 0(e~Sw/8) = o A) if m = [S]. Also
1
V
1 —
P
,1+*
V
Hence,
or
Now
by
C.15.2),
log!
login- ^
=
1 ?A+^I
; (i+€)iogi
- 2 log 1-^1+0A)
p ^ ttr * "'
loglogt(r+y+o(l),
= (l+€){2i8(a) + '?}loglog^,

14.9 CONSEQUENCES OF RIEMANN HYPOTHESIS 347
and taking a arbitrarily near to 1, we obtain A4.9.1). Similarly, by
C.15.3), l
= loglog Ttr+log—- + o A),
and A4.9.2) follows from this.
Comparing Theorem 14.9 with Theorems 8.9 (A) and (B), we see that,
since we know only that j3(l) < 1, in each problem a factor 2 remains
in doubt. It is possible that j3(l) = \, and if this were so each constant
would be determined exactly.
14.10. The function 8{t). We shall next discuss the behaviour of
this function on the Riemann hypothesis.
If I < a < a < 0, T < t < T', we have
, B + iT' oc + iT' <x + iT B + iTK
**™=MI+ S+1+ J
B + iT B + iT' OL + iT' <x + iT
Let 0 > 2. By A4.2.2),
2 + iT
ol + xT v a
B + iT B+iT
r log^(z) ^ f n~z
Also I dz = y Ai(n) j dz.
2 + iT n=% 2 + iT
Now
I >7 I '— I 1/7
J 2 — 5 Z ~~ [B —5)l0g7lJ lOgn J (Z — 5J
2 + iT + J 2 + iT
= ^^(^(^y))^2 J (x-oJ+(t-TJ] = O[n2(t-T)}'
^ —00
j3+iT
Hence — dz = 01-——I,
«/ \ /
2 + iT
and hence
T
Z—5 \f-
uniformly v/ith respect to ?. Similarly for the integral over

348 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
B + iT'
Also f
Making /? -> oo, it follows that
A similar argument shows that, if R(s') < \,
OL + iT
i
Taking s' = 2a—a-{-it, so that
s'—z = 2oc—o-{-it—((x-{-iy) = a—iy—(o—it),
and replacing A4.10.2) by its conjugate, we have
OL + iT'
i
A4.10.3)
From A4.10.1) and A4.10.3) it follows that
i
and
i J ^
a + iT
14.11. We can now show that each of the functions
max{log |?(«)|,0}, max{-log K(«)|,0},
max{arg ^E), 0}, max{—arg ^E), 0}
has the same v-function as log ?(s). Consider, for example,
max{arg t,(s), 0},
and let its v-function be vx(a). Since
I arg ?(*)(<
we have at once M0") ^

14.11 CONSEQUENCES OF RIEMANN HYPOTHESIS 349
Also A4.10.5) gives
T'
arg4E) = -
A4.11.1)
<
taking, for example, T = \t, T' = 2t.
It is clear from this that i/x(ct) is non-increasing. Also the Borel-
Carath^odory inequality, applied to circles with centre 2-{-it and radii
2—a—8, 2—a—28, gives
If a+8 < 1, so that v(a+8) > 0, it follows that
Since e and 8 may be as small as we please, and v(a) is continuous, it
follows that 1 \ ^ , \
v{oc) < ^(a).
Hence vx{a) = v{a) (| < a < 1).
Similarly all the v-functions are equal.
14.12. O.-results^ for S(t) andS^t).
Theorem 14.12 (A). Each of the inequalities
S(t) > (log*)i-e, A4.12.1)
S{t) < -(\ogt)-2~' A4.12.2)
has solutions for arbitrarily large values of t.
Making a -> \ in A4.11.1), by bounded convergence
? A4.12.3)
If S(t) < logat for all large t, this gives
2*
?
¦f Landau A), Bohr and Landau C), Littlewood E).

350 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
The above analysis shows that this is false if a < v{o), which is satisfied
if a < \ and a is near enough to \. This proves the first result, and the
other may be proved similarly.
Theorem 14.12 (B).
From A4.10.5) with a -> | we have
2*
log ?(«) = » f S[y). dy+O(l)
J s—i—W
It
,2*
2*
a,
i
since S^y) = O(\ogy). The result now follows as before.
In view of the result of Selberg stated in § 9.9, this theorem is true
independently of the Riemann hypothesis. In the case of S(t), Selberg's
method gives only an index ? instead of the index \ obtained on the
Riemann hypothesis.
14.13. We now turn to results of the opposite kind.f We know that
without any hypothesis
8(t) = O(logt), S^t) = O(\ogt),
and that on the Lindelof hypothesis, and a fortiori on the Riemann
hypothesis, each 0 can be replaced by o. On the Riemann hypothesis we
should expect something more precise. The result actually obtained is
Theorem 14.13. ,
(
We first prove three lemmas.
Lemma a. Let
= max
50 that <j>{t) is non-decreasing, and <j>(t) = O(log^). Then
8(t) = 0[{4>Bt)\Qgt}*}.
t Landau A1), CramSr A), Littlewood D), Titchmarsh C).

14.13 CONSEQUENCES OF RIEMANN HYPOTHESIS 351
This is independent of the Riemann hypothesis. We have
N(t) = L(t) + R(t),
where L(t) is denned by (9.3.1), and R(t) = S(t) + O(l/t). Now
"¦U N(T+x)-N{T) > 0 @ < x < T).
Hence
R(T+x)-R(T) > -{L(T+x)-L(T)} > -AxlogT.
Hence T+x
J R(t)dt = xR(T)+j {R(T+u)-R{T)}du
o
x
Hence T+x
> xR(T)—A ( ulogT du
o
>xR(T)—Ax2log T.
T + x
R(T) <- f R(t)dt+Ax\ogT
a;
log
Taking x = {<?BT)/log T}!, the upper bound for S(T) foUows. Similarly
by considering integrals over (T—x, T) we obtain the lower bound.
Lemma j3. Let a < 1, a?zd let
Then
We apply Hadamard's three-circles theorem as in § 14.2, but now take
We obtain M2 < AM% = AMz{\jMzy-a,
where M3 < F(T+1),

352 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Hence
M2 < AF( T+ i)i-A(*-i-4) ^ AF( T+ 1 ^
This gives the required result if a ^ f, and for f ^ a ^ 2 it is trivial,
if the A is small enough.
Lemma y. For a > \, 0 < ? < \t,
log?(«) = * f y. dy+o(^Pl+Q(i)- (H.13.3)
We have
«2«I f
= O(— ¦
and similarly for the integral over (%t,t—?). The result therefore follows
from A4.12.4).
Proof of Theorem 14.13. By Lemmas a and y,
t+i
for a-\ > 1/loglog T, 4 < < < T. Taking
log*/ (lo
we obtain log C(«) = O[(log !T)l(loglog T)
Hence by Lemma ]8, for ct—| > 1/loglog T,
log ^E) =
Hence
J log | ?(«) | rfa = O[(log T)^(lo
i+i/iogiog r
' A4.13.4)

14.13 CONSEQUENCES OF RIEMANN HYPOTHESIS 353
Again, the real part of A4.13.3) may be written
Hence
J log | ?(«) | da = f arctan ^ {?(*-*)-?(*+*
\ o *
or
=
Varying T and taking the maximum,
= J (^
A4.13.5)
Taking //. = 1/loglog T, and ^ as before,
? + l/loglogT
J log | C(s) I ^ = O[(log T)^(lo
A4.13.6)
Now A4.13.4), A4.13.6), and Theorem 9.9 give
Stf) = O[(log T)^(loglog T)-i{4>ET)}l] D < « < T). A4.13.7)
Varying ^ and taking the maximum,
4>(T) = 0[(log r)i(loglog r)-
Let i(T) = max
so that *p{T) is non-decreasing and
Then A4.13.7) gives
But *Jj{5Tx) < 60B^) for some arbitrarily large 2^; for otherwise
i.e. 0(T) > AT for some arbitrarily large T, which is not so, since in
fact 4>{T) = O(logT), ifj(T) = O{(loglog TJ}. Hence

354 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
for some arbitrarily large Tlf and so for all Tx, since «/> is non-decreasing.
Hence <f>(T) = o[- l°gT ).
9K J l(loglogT)*/
This proves A4.13.2), and A4.13.1) then follows from Lemma a.
The argument can be extended to show that, if Sn(t) is the nth integral
of S(t), then .
14.14. Theorem 14.13 also enables us to prove inequalities for ?(s)
in the immediate neighbourhood of a = |, a region not touched by
previous arguments. We obtain first
Theorem 14.14 (A).
We have
S(t+x)-8(t) = {N(t+x)-N(t)}-{L(t+x)-L(t)}-{f(t+x)-f(t)},
where/(^) is the O(l/t) of (9.3.2), and arises from the asymptotic formula
for logl». Thus/'@ = O(l/t2), and since N(t+x) > N(t)
S(t+x) — S(t) > —Axlogt+O(x/t2) > — Axlogt.
Hence, by A4.13.5),
uniformly for a > \, and so by continuity for a — \. Taking
^= 1/loglog*
the result follows.
Theorem 14.14 (B). We have
2 ) . ,rAlogt
(i < cr < i +^/loglogO, A4.14.2)
arg?(s) = 0\ 6 (| < a < 1+^4/loglogO- A4.14.3)
\loglog</
By A4.13.1) and A4.13.3),

14.14 CONSEQUENCES OF RIEMANN HYPOTHESIS 355
? f/(a-i)
m f du _ C dx'
J V((--iJ+*2}= J V0+^)'
0 0
which is less than 1 if ? ^ ct—^, and otherwise is less than
f
J
cr—
Taking ? = 1/loglogi, the lower bound in A4.14.2) follows. The upper
bound follows from the argument of the previous section. Lastly,
taking imaginary parts in A4.13.3),
NOW 2 ^ 1X2 dX < I 2 ^ 1X2 ^ = 2^'
J Z2+(<7--?J J X2+(a— ^J
0 0
Hence, taking ? = 1, A4.14.3) follows uniformly for cr > ?, and so by
continuity for a = ^.
In particular
(l \ ( + f V A4.14.4)
Uoglogi/
From A4.14.4), A4.5.2), and a Phragmen-Lindelof argument it
follows that fflncr/>22a\
1 A
uniformly for -+ -—= < a < 1—8.
2 loglogi
14.15. Another result in the same order of ideas is an approximate
formula for log ?(s), which should be compared with Theorem 9.6 (B).
Theorem 14.15. For !< a < 2,
log Us) =
I
K-yKl/loglog*
A4.15.1)
where p = \-\-iy runs through zeros of ?(s).

356 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
In Lemma a of § 3.9, let
where 8 = 1/loglog T. By A4.14.4)
1
The upper bound in A4.14.2) gives
for 15—50| < r, a > |; and for \s—so\ < r, a < -|, the functional equa-
equation gives
T) r\loglog Tj
It therefore follows from C.9.1) that
log ?(s)—log ?(s0)— 2 logE—
|so-pK28/V3
for 15—50| < |r, and so in particular for \ < a
Also So-p = _L8+i(T-y).
Hence V3§ ^ l5o~Pi < A>
and so, if the logarithm has its principal value,
log(*o-/>) = °(log^) = O(\og\og\ogT).
Also the number of values of /> in the above sums does not exceed
by Theorem 14.13. Hence
1<

14.15 CONSEQUENCES OF RIEMANN HYPOTHESIS 357
Since \T—y\ < S if \so—p\ < 2S/V3, the result follows, with T for t
and ?^cr<?+S. It is also true for \-\-h < a < 2, since in this region
= °(loglog t}
and, as in the case of the other sum,
This proves the theorem.
For ?'(s)/?(s) we obtain similarly from Lemma a of § 3.9
I*-yl^ 1/loglog* H
A4.15.2)
14.16. Theorem 14.16. Each interval [T, T+l] contains a value of t
such that / loe? \
K(s)|>expf -A^^J (}<a<2). A4.16.1)
Let <5 = 1/loglog T. Then the lower bound A4.16.1) holds automati-
automatically for a ^ \ + <5, by A4.14.4). We therefore assume that \ ^ c ^ ? + <5.
If s = o- + it and s0 = \ + 3 + it then, on integrating A4.15.2), we find
s-p\ i nl logT ^
Moreover logC(s0) = O( -— — ) by A4.14.4) so that, on taking real
—4... \loglog T)
parts
loglC(«)l= I log
s-p
+ 01
Jog_T_N
loglog T
since \s-p\ > \t-y\ and \so-p\ < 23. We now observe that
T
j \t-
T-d^y^T+1+5 J
(-2<5-2<51og2)
+S
-AdlogT,

358 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
as there are O(log T) terms in the sum. Hence there is a t for which
and the result follows.
In particular, if e is any positive number, each (T, T-\-l) contains a
, t such that 1
= 0{te) (|< a < 2). A4.16.2)
14.17. Mean-value theoremst for S(t) and S^t). We consider first
^t). We begin by proving
Theorem 14.17. For \T < * < T, S = T~i,
loglog
) A4J7-1)
00
where C = I log | ?,(<j) \ da.
2
Making /? -> oo in (9.9.4), we have
00
-TTiS^O = C— j log|C(or+iOI rfa. A4.17.2)
Now, integrating A4.4.1) from 5 to \-\-it,
iogCM-]»gC<|+a> =
n=2 n=2
H 1)
f 8«>-Pr(p-5l) da. + O^-Hogt) (i < a
Also, if a ^ |,
2^
n=2 n=2
= O{(da-1+2-a5)\ogt}. A4.17,3)
Hence, for \ < a ^ |,
n = 2
f Littlewood E), Titchmarsh B).

14.17 CONSEQUENCES OF RIEMANN HYPOTHESIS 359
and integrating over \ ^ a ^ |,
5. 9. 9
8 8 m 8"
I log?(s) c?ct = I ( / ———-e~°n\do-\- / I (a-,—^)8Sl~^r(p—S,
i i v-« y •J-
Also, by A4.17.3),
oo
J log C(S) da =
00 00
/ oo
and
II
n"
the inversion being justified by absolute convergence. Hence
= C- y^Wcostflogn)^;
^ n^ log n
.9
8
r / _
Now tv _ \
(ly-«l > i)>
Hence
1= I + I +
|<|<l/ll< 1/Il«|<|<1
|y-<|<l/loglog< J

360 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Now f
o
f (^-i^-i ^ < | xe-xlogl/8 «& = lOg-2(l/8).
As in §14.5,
l
Also, by A4.13.1), for *— 1 < t' <
Nlt + ^)
\ ^loglogt)
Hence I 1 =
|y -<K 1/loglog < \l0gl0g «/
and
y —- y t —
Z^ V —^ ^w Z^ y — t
t+1/loglogt<y^t + l ' m<loglog< <+m/loglogKy« + (m+l)/loglog< r
m<loglo
Hence
V of—^-1 _ lo^ \ = 0(log«logloglogO.
A_# WloglogHoglog*/ 6 g g g;
y = 0/ lQg^ \ , oAog^ogloglogA . 0(
f \logl/81oglog«r I log21/3 /"*"
( Q(
Uog2l/8/ \\og\ogt)
for the given 8 and t. This proves the theorem.
14.18. Lemma 14.18. If an = 0A), 8 < \, then
5
uniformly for a > \. Similarly, if an = O(\ogn), the formula holds with
a remainder term , , ,v
The left-hand side is
T
00 00
W
oo , ,2
Clearly ? = V 1M. e~^n.
m = n ^—i n
n=l
Also 2 = 0[I ^
m<n

14.18 CONSEQUENCES OF RIEMANN HYPOTHESIS 361
Now
y —4 = o
-m+i--' --¦-'- n.=»+i------ ^-w)/™w
Bm , v
g-m8 V \ _ O(e-m8logm),
Zw n—ml
n = m + l '
n = m + l
-nB
1 {mn)logn/m \m /-1 I \m8 I'
Hence
This proves the first part. In the second part we have a pair of
logarithms running throughout the remainder terms, and this is easily
seen to produce the extra log21/8 in the result.
14.19. Theorem 14.19. AsT-+oo,
n~2
Let f(t) = Ce-L f A'(M)c,°,s(<log").-«-
6
n = 2
Then, as in the lemma,
and we can replace 8 by 0 in the first two terms on the right with error
\ /
+0
Hence, taking 8 = T-i,
T
2
Hence
T T
2tt2 r
T
= ! J w^+ofen^ I i/wi

362 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
and, since J \f(t)\dt < J dt J {/(*)}2 d« a =
it follows that
T
J
Replacing T by \T, \T,... and adding, we obtain the result.
14.20. The corresponding problem involving S(t) is naturally more
difficult, but it has recently been solved by A. Selberg D). The solution
depends on the following formula for ?'(
Theorem 14.20. Without any hypothesis
ns A— sJlog
l
x ^
g=l
where
Let a = maxB,1+cr). Then
a+x'oo
Xl-!- dz
-¦ [
2tti J
(z-sf
a—xoo
oo a + i0°
1 - v !* r~Z—S
a—xoo
00
_J_yAW r
27ri -^ n? J
a— a— xoo
a —cr + xoo a —cr + ioo
f (fV—- f tT-
J U/ ™2 J U/ H
—cr—xoo a — a— xoo
ns \ &n *n /-, n

14.20
CONSEQUENCES OF RIEMANN HYPOTHESIS
363
Now consider the residues obtained by moving the line of integration
to the left. The residue at z = s is — logx?'(s)/?(s); that at z = 1 is
Xl-S_x2(l-s)
those at z = —2q and z = p are
g.-2q—s #2(—2q— s)
respectively. The result now easily follows.
14.21. Theorem 14.21. For t > 2, 4 < x
t2,
have
= l V
it Z-,
n<x3
By the previous theorem,
O<
nai log n (log x
n<x3
for a ^ o-j, where |a>| < 1. Now
-v-2(l — a) _i_ -v-1 —a
, p
\ogxj
A4.21.1)
Hence
n<x
y
A4.21.2)
Now by B.12.7)
T.f—
Hence

364 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Taking real parts in A4.21.2), substituting this on the left, and taking
a = crlf
= -R
|
n^+it ' e Z-,{ax —
Hence
Ax{n)
Here
Hence
, 2to' . 2 1
1 > 1 > -.
e e 4
= 0
n<x2
Inserting this in A4.21.2), we get
^f + OJxi-*7 V —
(o+it)
i
n<x2
Now
By A4.21.4)
CO
2
=
A4.21.4)
<7i
I ^ I ~Y ~ '•—
Cf(g+*O
o
n<x
wff>+aloew I log a;
Ax(n)
n<x2
n<7i+i<
\ o/logA
I Uoga;/

14.21 CONSEQUENCES OF RIEMANN HYPOTHESIS
Also, by A4.21.4) with a = av
365
n<x2
noi+it
n<x2 ' \ o /
It remains to estimate Jz. For \ < a < cr1
+ 0(logO.
Hence
?(o+it)\
+ 0(logt).
Hence
<
? C^fei J
logo:
2
n<x2
by A4.21.3). The theorem follows from these results.
Theorem 14.21 leads to an alternative proof of Theorem 14.13; for
taking x = ^/(log t) we obtain
8(t) =
loglog

366 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
14.22. Theorem 14.22. For
Ta < a; < T* @ < a < \)
± y
We have
1 y analogy) = I y
P<X2
¦ sinolog jp) +
+olA
(
liogx
V 6
2 -r
y
The last term is bounded if^T<^< T, x ^ To, where a is a fixed
positive constant. The last term but one is
Now consider the first term on the right. If p < x,
ai = A— ^-
and, if a; < p ^ x2, it is
Hence the first term is the imaginary part of
where
Now
ap =
\ogp
r
*
*)*

14.22 CONSEQUENCES OF RIEMANN HYPOTHESIS 367
Since 4 = O
— V l^\ = O(T),
the first term is oI
by C.14.3). The second term is
0{ y y ^M+ol y y
= of y -t&z-Y+ol y
2)= 0{T)
if x < VT.
A similar argument clearly applies to the second term. In the third
term, the sum is of the form
where ocp = 0A); and
f
V
Z,
p<x
JT
= °v
) = 0(T).
Similarly for the fourth term, and the result follows.
14.23. Theorem 14.23. // Ta < x < Tt,
f
J
p<x2

368 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
This is
T
>> >> —r-i sin(tlog p)sin (t log a) dt
p<x* q<x*r * gT
Now, by C.14.5),
= loglogT+0(l)
and (since pn > An log n)
Y-I—= 0A).
^plogp
Hence the first term is
iT\og\ogT+O(T).
Also \logp/q\ > A\p > A\x2.
Hence the remainder is
01 x21 "V —I I — Hlv2 ^\ —
and the result follows if x
14.24. Theorem 14.24.
T
For
T T
1 -ST si
T
'1
T
77 J
T
+4f(^
= O{T)+O{T{\og\ogT)\} + ^-2T\og\ogT+O{T)

14.24 CONSEQUENCES OF RIEMANN HYPOTHESIS 369
(using Schwarz's inequality on the middle term). The result then
follows by addition.
It can be proved in a similar way that
T
J {S(t)?* dt ~ ^^ T(loglog Tf A4.24.1)
o
for every positive integer k.
14.25. The Dirichlet series for l/?(s). It was proved in § 3.13 that
the formula , »
which is elementary for a > 1, holds also for a = 1. On the Riemann
hypothesis we can go much farther than this.f
Theorem 14.25 (A). The series
^ A4.25.1)
ns
n = l
is convergent, and its sum is l/?(s), for every s with a > \.
In the lemma of § 3.12, take an = fx(n), f(s) = l/?(s), c = 2, and x
half an odd integer. We obtain
2+iT
^~^i J t,{s+w) ~w
n J ,{+) \T
n<x J bV ' v
-o + 8-iT \-o + 8+iT 2 + iT
o + 8+iT 2 + iT .
G + 8-iT ^-cr + S + iT'
2-iT ? ^
where 0 < 8 < ct—|.
By A4.2.5), the first and third integrals are
o
t-1j" f x" du\ =
and the second integral is
/ T
0 ari-'+* [
| Littlewood A).

370 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Hence V
Taking, for example, T = x3, the 0-terms tend to zero as x -> oo, and
the result follows.
Conversely, if A4.25.1) is convergent for a > \, it is uniformly con-
convergent for a > ct0 > \, and so in this region represents an analytic
function, which is l/?(s) for a > 1 and so throughout the region. Hence
the Riemann hypothesis is true. We have in fact
Theorem 14.25 (B). The convergence of A4.25.1) for a > \ is a
necessary and sufficient condition for the truth of the Riemann hypothesis.
We shall write
M{x) =
Then we also have
Theorem 14.25 (C). A necessary and sufficient condition for the
Riemann hypothesis is M^ = 0{x\+*). A4.25.2)
The lemma of § 3.12 with s = 0, x half an odd integer, gives
2+iT
A4.25.3)
T
= ol J \
-T
by A4.2.6). Taking T = x2, A4.25.2) follows if x is half an odd integer,
and so generally.
Conversely, if A4.25.2) holds, then by partial summation A4.25J)
converges for a > \, and the Riemann hypothesis follows.
14.26. The finer theory of M(x) is extremely obscure, and the results
are not nearly so precise as the corresponding ones in the prime-number
problem. The best 0-result known is

14.26 CONSEQUENCES OF RIEMANN HYPOTHESIS
Theorem 14.26 (A).f
371
M{x) = 0\x\
To prove this, take
\ loglog xjj
A4.26.1)
1
loglog T'
in the formula A4.25.3). By A4.14.2),
T = x\
loglog TJ
on the horizontal sides of the contour. The contribution of these is
therefore
4 iog* \\
loglog xj J
On the vertical side, A4.14.2) gives
1
\ loglog v loglog v J
for v0 < v < T. Now it is easily seen that the right-hand side is a
steadily increasing function of v in this interval. Hence
Hence the integral along the vertical side is of the form
loglog TJ
J v
This proves the theorem.
Theorem 14.26 (B).
A4.26.2)
This is true without any hypothesis. For if the Riemann hypothesis
is false, Theorem 14.25 (C) shows that
M{x) =
| Landau A3), Titchmarsh C).

372 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
with some a greater than \. On the other hand, if the Riemann hypo-
hypothesis is true, then for a > \
OO-2, n- -nZM{n)W (n+iy}-S)
«*) ..1
Suppose that A4.26.3)
\M(x)\ ^ i?0 A < x < x0), ^ Szi (x > x0).
Then
x0
S \Mn —?
00 00
/ /7/y I /I'V
+ J^. A4.26.4)
But if p = ?+iy is a simple zero of ?(s), and s = o-\-iy, a -> \, then
1 1 1
Us) Us)
We therefore obtain a contradiction if
8<
This proves the theorem.
14.27. Formulae connecting the functions of prime-number theory
with series of the form p
2
7
etc., are well known, and are discussed in the books of Landau and
Ingham. Here we prove a similar formula for the function M(x).
Theorem 14.27. There is a sequence Tv, v < Tv < v+1, such that
n=l ()
i/ x is noi an integer. If x is an integer, M(x) is to be replaced by
In writing the series we have supposed for simplicity that all the
zeros of ?(s) are simple; obvious modifications are required if this is not so.

14.27 CONSEQUENCES OF RIEMANN HYPOTHESIS 373
For a fixed non-integral x, C.12.1), with an = fx(n), s = 0, c = 2, and
w replaced by s, gives
2 + iT
2tti J s ?(s) \T]
2-iT
If x is an integer, ^(x) is to be subtracted from the left-hand side.
By the calculus of residues, the first term on the right is equal to
T7~\ * \
(-2N-1-iT ~2N-l + iT
J J J
2-iT -2iV-l-iT _2JV-l + i5
where T is not the ordinate of a zero. Now
-2N-1+IT 2N+2 + iT
J HiF) s=z J (i-^(i-.
2iV+2-iT
2iV+2+iT
J
.—s
Here
Hence the integral is
T
which tends to zero as N -> oo, for a fixed T. Hence we obtain
oo-iT 2 + iT
" 21 y (-i)BW^, i f ,
T
1r i /2tt\2-v+
Jtt\x)
Also
f
/» o /» 1 « (* ^A
I /» O ^— I /TO ^— I
I YI \ 1/1 \Y11 \ ll
¦oo+iT 2 + iT 2 + iT
oo
-il *$

374 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
Also by A4.16.2) we can choose T = Tv (p < Tv < v+1) such that
¦± = 0(f) (i < a < 2, t = Tv).
Hence for — 1 < a < \, t = Tv
2+iTv
Hence
f
J
1 + iT
Similarly for the integral over B—iT, — oo—iT), and the result stated
follows.
It follows from the above theorem that
1
?(p)\
is divergent; if it were convergent,
would be uniformly convergent over any finite interval, and M(x)
would be continuous.
14.28. The Mertens hypothesis.f It was conjectured by Mertens,
from numerical evidence, that
\M(n)\ < Vn (n> 1). A4.28.1)
This has not been proved or disproved. It implies the Riemann hypo-
hypothesis, but is not apparently a consequence of it. A slightly less precise
hypothesis would be M{x) = 0(x|) A4.28.2)
The problem has a certain similarity to that of the function ifj(x)—x
in prime-number theory, where
f{x) = 2 A(n).
On the Riemann hypothesis, ifj(x)—x = O(x2+e), but it is not of the
form O(x%), and in fact
ifj(x)—x = Q(xilogloglogx). A4.28.3)
The influence of the factor logloglogx is quite inappreciable as far as
f See references in Landau's Handbuck, and von Sterneck A).

14.28 CONSEQUENCES OF RIEMANN HYPOTHESIS 375
the calculations go, and it might be conjectured that A4.28.2) could be
disproved similarly. We shall show, however, that there is an essential
difference between the two problems, and that the proof of A4.28.3)
cannot be extended to the other case, at any rate in any obvious way.
The proof of A4.28.3) depends on the fact that the real part of
2eipz
y>0 *
is unbounded in the neighbourhood of z = 0. To deal with M(x) in the
same way, we should have to prove that the real part of
is unbounded in the neighbourhood of z = 0. This, however, is not the
case. For consider the integral
1 f pis
ni J S ?(
2ni J S Us)
taken round the rectangle ( — 1, 2, 2-\-iTn, —l+iTn), where the Tn are
those of the previous section, and an indentation is made above s = 0.
The integral along the upper side of the contour tends to 0 as n -> oo,
and we calculate that
2+ioo _ -1+ioo _ 2
f{z) = —. -^—ds . -^—ds-\ . e ds.
2 -1 -1
The last term tends to a finite limit as z -> 0. Also
\eUz\ = ey~xt < ev (s = —\-\-it, z = x-\-it, x > 0)
and l/?( —1+*?) = O(t~%). The second term is therefore bounded for
0.
The first term is equal to
ds.
J
sns
Now, if n > 1,
2y°°es(x3-logn) T es(iz-\ogn) -j2 + t°° \ f » j
J s S~ [s{iz—logn)\2 iz—\ogn J
2 + ioo „ . 2 + ioo
es(iz-\ogn) r es(iz-\ogn) n2+x« j r es(iz-\ogn)
|eB+«Xk-logn)| = e-2y~2\ogn-tx
2 + ioo
J
es(iz-\ogn)

376 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
uniformly in the neighbourhood of z = 0. Hence
2 + ioo
If z = re^, we have
2 + xoo e»(lir-fl)
J ?*- J + J
= 0A)+ J
00
^-dX
r
1
Hence f(z) = ^-. log- + 0A),
and consequently R/(z) is bounded.
14.29. In this section we shall investigate the consequences of the
hypothesis that x
f l^lYdx = O(logX). A4.29.1)
j \ x )
i
This is less drastic than the Mertens hypothesis, since it clearly follows
from A4.28.2). The corresponding formula with M{x) replaced by
ip{x)—x is a consequence of the Riemann hypothesis.f
Theorem 14.29 (A). // A4.29.1) is true, all the zeros of ?(«) on the
critical line are simple.
By A4.26.3),
00 00
r \M{x)\, , , r \M{x)\ i ,
-—— dx = \s\ is -i—i dx
I /v.ct+1 I o-a^+S" T'5(T+1
ICO CO CO
1 1 ; vl
| Cramer E).

14.29 CONSEQUENCES OF RIEMANN HYPOTHESIS 377
x
Let f(X) = f f^lYdx.
Then
00 00 00
dx
-r-dx = J-^?dx = (a— -i) ^-\
11
Hence JL = o( Ji
Let p be a zero and 5 = p-\-h, where h > 0. Then a = \-\-~h, and
hence ,
This would be false for h -> 0 if p were a zero of order higher than the
first, so that the result follows.
Multiplying each side of A4.29.2) by h, and making h -> 0, we obtain
J A*.29.3)
i KP)
We can, however, prove more than this.
Theoeem 14.29 (B). // A4.29.1) is true,
2 rw <14-29-4>
is convergent.
This follows from an argument of the 'Bessel's inequality' type. We
have
o ^
J
lrfe Pi (P)
l/l
(p) J

378
CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
In the first sum, the terms with p = \—p are
x
j
since 1 — p is the conjugate of p. In the remaining terms, p = \-\-iy.
p = \-\-iy', where y =?¦ —y. Hence
J
ly+/
Hence the sum of these terms is less than Kx = K-^T).
In the last sum we write
xx x
f M{x)xP~2 dx = [
^- f M{x)xP~1 dx.
The last term is
°
by A4.29.1). Also
f -2/ x\j 1 f
J \ XJ 2ni J
2 + ioo
- A4-29-5)
1 2—ioo
To prove this, insert the Dirichlet series for l/?,(w) on the right-hand
side and integrate term by term. This is justified by absolute con-
convergence. We obtain
oo 2 + ioo
Tew f i. xw+p'1-1
i
2—ioo
w(w+p)(w+p~l)
dw.
Evaluating the integral in the usual way by the calculus of residues,
we obtain
x
Xp—up\
. AXP-^—nP-1
/*» i
X
= J
2
> /Xl 7w III vt* — I (JLQC
n
X
siT I 1\/Tt^y*\/y*P—2/ 1 1 nT
iX<L — 1 lri yx)x~ 1 1 — 1 ux,
and A4.29.5) follows.

14.29 CONSEQUENCES OF RIEMANN HYPOTHESIS 379
Let U be not the ordinate of a zero. Then the right-hand side of
A4.29.5) is equal to
2-iU \-iU %+iU 2+iU
2tti\
x" " |"i J 2 + iU
+sum of residues in — U < I(w) < ?7.
Let p" run through zeros of ?(s) with imaginary parts between — U
and U. Let U > T. Then there is a pole at w; = I—p, with residue
At the other p" the residues are
y'( "\ 'f'Z'1"™ -j- i= °(rr^r—\\f »j_ m) = °L  \X
? [p )p [p +p—i)(p +p) Wip +p—}){p +p)\J \\y +y\ I
by A4.29.3), and
where iJT2 depends on T, if |y| < T, but not on U.
Again
2+ioo
f
f ^+p-1~1 dw = on f^
J ^^(w+pj^+p-l) \ J v(v
2 + iU V C/
)¦
\U(U-T)
and similarly for the integral over B—ico, 2—iU). Also by A4.2.6) and
the functional equation
Hence, since |w+pl ^ f> I^+P—1| ^ h
u
i + iU . U v
C xw+P-1 1 I f l-yl€~4
±— dw = Ol ' ' x^ = 0A).
Finally, by Theorem 14.16, we can choose a sequence of values of U
such that i
_i_ = 0(M) (<= U, ^a^2).

380 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
By the functional equation the same result then holds for ? ^ a ^ \
also. Hence
2 + iU
]
f
J
_ I Zt \ _ (
\c/i€(C/+JJ - (c)
J t,{w)w{w+P){w+p-
and similarly for the integral over B—iU, \—iU). Making U -> oo, it
follows that
where \R\ < A = K3(T) if \y\ < T.
Hence we obtain
y _i_-2iogz y
21
lyKT I/>?(/>) I-
Since the right-hand side is now independent of T, the result follows.
In particular ——= o{\p\).
14.30. // A4.29.1) is true,}
Suppose that the interval (f—^-3,i+i-3) contains y, the ordinate of
a zero. By differentiating B.1.4) twice,
Using this and A4.29.3), we obtain
V
y\
7
A_A A
> 1 72 > T'
Cramer and Landau A).

14.30
CONSEQUENCES OF RIEMANN HYPOTHESIS 381
3) is free from ordinates of
Suppose on the contrary that (t—i~3j
zeros. Theorem 14.15 gives
iog|«l+*)l= 2
There are O{logt/\og\ogt) terms in the sum, each being now
Hence
\\og\ogt)
Now tt-2sT(^s)^(s) is real on a = \. Hence
is purely imaginary on a = \. Hence, on a = \,
Hence (without any hypothesis)
(t>t0).
This proves the theorem.
14.31. Let \-\-iy, \-\-iy' be consecutive complex zeros of ?($). If
A4.29.1) is true
, A / , logy \
y'—y> -exp — A-—f-*—\.
We have
0 = J ^(i+i<) * = (y/-
J (y'—or(J
Hence by A4.29.3)
y'
y~y <a\
<A max
Now
f (y'-0 ^ = ^(y'-
max
}^dg=
2tt

382 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
by A4.14.1) and the functional equation. Taking r =
\ \
and the result follows.
14.32. Necessary and sufficient conditions for the Riemann hypothesis.
Two such conditions have been given in § 14.25. Other similar con-
conditions occur in the prime-number problem.f
A different kind of condition was stated by M. Riesz.J Let
A4-32J)
Then a simple application of the calculus of residues gives
o+ioo o + ioo
[
a—ioo a—ioo
where \ < a < 1. Taking a just greater than ?, it clearly follows that
F{x) = O(xi+<).
On the Riemann hypothesis we could move the line of integration to
a = ^-f e (using A4.2.5)) and obtain similarly
F{x) = O(zs+€). A4.32.2)
Conversely, by Mellin's inversion formula,
00
?B*)
= - f F(x)x-1's ds.
If A4.32.2) holds, the integral converges uniformly for a > ct0 > ?; the
analytic function represented is therefore regular for a > \, and the
truth of the Riemann hypothesis follows. Hence A4.32.2) is a necessary
and sufficient condition for the Riemann hypothesis.
A similar condition stated by Hardy and Littlewood§ is
Jr7oJl^ A4.32.3)
Z,fc!?B&H-l)
These conditions have a superficial attractiveness since they depend
explicitly only on values taken by ?(s) at points in a > 1; but actually
no use has ever been made of them.
f Landau, Vorlesungen, ii. 108-56. % M. Riesz A).
§ Hardy and Littlewood B).

14.32 CONSEQUENCES OF RIEMANN HYPOTHESIS 383
Conditions for the Riemann hypothesis also occur in the theory of
Farey series. Let the fractions h/k with 0 < h ^ k, (h, k) = 1, k ^ N,
arranged in order of magnitude, be denoted by rv (v = 1, 2,..., <5>(N),
where <b(N) = <f>(l) + ...+<j>{N)). Let
K = rv-v/<&(N)
be the distance between rv and the corresponding fraction obtained by
dividing up the interval @,1) into O(iV) equal parts. Then a necessary
and sufficient condition for the Riemann hypothesis isj
An alternative necessary and sufficient condition is§
<). A4.32.5)
v=l
Details are given in Landau's Vorlesungen, ii. 167-77.
Still another condition|| can be expressed in terms of the formulae of
§ 10.1. If E(t) and <X>(w) are related by A0.1.3), a necessary and sufficient
condition that all the zeros of E(t) should be real is that
00 00
J J 0(a)OOS)e*a+/fce(a-#V-jSJ dadfi > 0 A4.32.6)
— 00 —00
for all real values of x and y. But no method has been suggested of
showing whether such criteria are satisfied or not.
A sufficient conditionff for the Riemann hypothesis is that the partial
n
sums 2 v~s °f tne series for ?(s) should have no zeros in a > 1.
NOTES FOR CHAPTER 14
14.33. The argument of §14.5 may be extended to the strip \ < cr0 < a
< f, giving
The choice b = (logi) then yields
C(s)
<
Us) l-a
\ Franel A). § Landau A6). || See Polya C), § 7. ft Turan C).

384 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
uniformly for <j0 < a < | and t ^ 2, and hence
log if 1+9
, ri , ^ <^ loglogi
log C(s) ^ <
(Iog01 1
— + logloglog t if <T0 < <7 4 1 + .
A - a) loglog * b b ° loglogf
These results, together with those of § 14.14 are the sharpest conditional
order-estimates available at present.
14.34. The O-result given by Theorem 14.12(A) has been sharpened by
Montgomery [3], to give
on the Riemann hypothesis. A minor modification of his method also
yields
It may be conjectured that these are best possible.
Mueller [2] has shown, on the Riemann hypothesis, that if c
is a suitable constant, then S(t) changes sign in any interval
[T, T + c loglog T].
Further results and conjectures on the vertical distribution of the
zeros are given by Montgomery [2], who investigated the pair corre-
correlation function
where w(u) = 4/D + u2). This is a real-valued, even, non-negative
function of a, and satisfies
F(a, T) = a + T-2* log T+ O(-i-) + O(aT—1) + O(T ~^)
VogTJ
A4.34.1)
for a $5 0, whence F(a, T) -> a as T-> oo, uniformly for 0 < S ^ a. ^ 1 — S.
Montgomery conjectured that in general
F(a, T) - min (a, 1) A4.34.2)

14.34 CONSEQUENCES OF RIEMANN HYPOTHESIS 385
uniformly for 0 ^ S ^ a ^ A. This is related to a number of conjectures
on the distribution of prime numbers. (See Gallagher and Mueller [1 ],
Heath-Brown [10], and joint work of Goldston and Montgomery in the
course of publication.) From A4.34.2) one may deduce that
logT ^' ' ^logT
for fixed a, ft, as T-> oo. Here d(a, fi) = 1 or 0 according as a ^ 0 ^ /? or
not.
Using A4.34.1), Montgomery showed that
where m(p) is the multiplicity of p, and Z' counts zeros without regard to
multiplicity. One may deduce, in the notation of § 10.29, that
N»(T) ^ {§¦ + o(l)}N(T), A4.34.3)
on the Riemann hypothesis. The conjecture A4.34.2) would indeed yield
NA)(T) ~ N(T), i.e. 'almost all' the zeros would be simple. Montgomery
also used A4.34.1) to show that
lim inf /^+/11~7" < 0-68; A4.34.4)
„->«* B7r/logyn)
here 27r/log yn is the average spacing between zeros.
By using a different method, Conrey, Ghosh, and Gonek (in work in
the course of publication) have improved A4.34.3). Their starting point
is the observation that
0<y< T
A4.34.5)
by Cauchy's inequality. The function M(s) is taken to be a mollifier
where the polynomial P(x) is chosen optimally as fa; — \x2. One may

386 CONSEQUENCES OF RIEMANN HYPOTHESIS Chap. XIV
write the sums occurring in A4.34.5) as integrals
and 2nl J C(s)
M(s)M(l-s)t'(s)t'(l-s)ds,
taken around an appropriate rectangular path P. The estimation of
these is long and complicated, but leads ultimately to the lower bound
The estimate A4.34.4) has also been improved, firstly by Montgomery
and Odlyzko [1], and then by Conrey, Ghosh and Gonek [1]. The latter
work produces the constant 05172. The corresponding lower bound
lim sup 7n + l~yn ^X>1 A4.34.6)
has been considered by Mueller [1], as well as in the two papers just
cited. Here the best result known is that of Conrey, Ghosh, and Gonek
[1], which has X = 2-337. Indeed, further work by Conrey, Ghosh, and
Gonek, which is in the course of publication at the time of writing,
yields X = 2-68 subject to the generalized Riemann hypothesis (i.e. a
Riemann hypothesis for ?(s) and all Dirichlet L-functions L(s, /).)
Moreover it seems likely that this condition may be relaxed to the
ordinary Riemann hypothesis with further work.
If one asks for bounds of the form A4.34.4) and A4.34.6) which are
satisfied by a positive proportion of zeros (as in § 9.25) then one may take
constants 0-77 and 133 (Conrey, Ghosh, Goldston, Gonek, and Heath-
Brown [1]).
14.35. It should be remarked in connection with § 14.24 that Selberg
D) proved Theorem 14.24 with error term O(T), while the method here
yields only O{T(loglog T)i}. Moreover he obtained the error term
for A4.24.1).
14.36. The argument of the final paragraph of §14.27 may be
quantified, and then yields
uniformly for T ^ To, assuming the Riemann hypothesis and that all the
zeros are simple. However a slightly better result comes from combining

14.36 CONSEQUENCES OF RIEMANN HYPOTHESIS 387
the asymptotic formula
Z IC(i
of Gonek [2] with the bound A4.34.3). Using Holder's inequality one
may then derive the estimate
y* —-—> t
lC/(i + iy)r '
where Z* counts simple zero only, and c > 0 is a suitable numerical
constant.
14.37. The Mertens hypothesis has been disproved by Odlyzko and
te Riele [1 ], who showed that
and
M(x)
hm sup —7-^- > 1-06
liminf—V^ < -1-009.
Their treatment is indirect, and produces no specific x for which
|M(jc)| > xk The method used is computational, and depends on solving
numerically the inequalities occurring in Kronecker's theorem, so as to
make the first few terms of A4.27.1) pull in the same direction. To this
extent Odlyzko and te Riele follow the earlier work of Jurkat and
Peyerimhoff [1], but they use a much more efficient algorithm for
solving the Diophantine approximation problem.
14.38. Turan C) conjectured that
n
A4.38.1)
for all x > 0, where X{n) is the Liouville function, given by A.2.11).
He showed that his condition, given in §14.32, implies the above
conjecture, which in turn implies the Riemann hypothesis. However
Haselgrove [2 ] proved that A4.38.1) is false in general, thereby showing
that Turan's condition does not hold. Later Spira [1] found by
calculation that
19
Z n-
n= 1
has a zero in the region a > 1.

XV
CALCULATIONS RELATING TO THE ZEROS
15.1. It is possible to verify by means of calculation that all the
complex zeros of ?(s) up to a certain point lie exactly (not merely
approximately) on the critical line. As a simple example we shall find
roughly the position of the first complex zero in the upper half-plane,
and show that it lies on the critical line.
We consider the function Z(t) = ei&t,(^-{-it) defined in § 4.17. This is
real for real values of t, so that, if Z{t^) and Z(t2) have opposite signs,
Z(t) vanishes between tx and t2, and so ?(s) has a zero on the critical
line between ^+i^ and \-\-it2.
It follows from B.2.1) that ?(?) < 0, then from B.1.12) that ?(?) > 0,
i.e. that E@) > 0; and then from D.17.3) that Z@) < 0.
We shall next consider the value t = 6tt. Now the argument of § 4.14
shows that, if x is half an odd integer,
Hence, taking t > 0,
cos(*logn-&) ^x? , -~-_ A5.1.2)
n<x
712
t
For ? = §,? = 6tt, the right-hand side is about 0-6.
We next require an approximation to &. We have
so that 0 = _?iog7T+Uogr{i+fr
It may be verified that the term 0A ft) is negligible in the calculations.
Writing & = 2nK, and using the values
log 2 = 0-6931, log 3 = 1-0986,
it is found that
K = 01166, 31og3-iC = 3-179
31og2-Z = 1-963, 31og4-Z = 4-042,

15.1 CALCULATIONS RELATING TO THE ZEROS 389
approximately. Hence the cosines in A5.1.2) are all positive, and
cos 2ttK = 0-74.... Hence ZFtt) > 0.
There is therefore one zero at least on the critical line between t = 0
and t = 6tt.
Again, the formulae of § 9.3 give
77
where A denotes variation along B, 2-\-iT, %-\-iT). Now R?(s) > 0 on
a = 2, and an argument similar to that already used, but depending on
A5.1.1), shows that R?(s) > 0 on B+iT,%+iT), if T = 6tt. Hence
|Aarg?(s)| < \-n, and
IVFtt) < f+2Z < 2.
Hence there is at most one complex zero with imaginary part less than
6tt, and so in fact just one, namely the one on the critical line.
15.2. It is plain that the above process can be continued as long as
the appropriate changes of sign of the function Z(t) occur. Defining
K = K(t), as before, let tv be such that
K{tv) = \v-l {v = 1, 2,...). A5.2.1)
Then A5.1.2) gives
cos(^logn)
n<x
If the sum is dominated by its first term, it is positive, and so Z(tv) has
the sign of ( — l)v. If this is true for v and v+1, Z(t) has a zero in the
interval (tv, tv+1).
The value t = 6?r in the above argument is a rough approximation
to t2.
The ordinates of the first six zeros are
14-13, 21-02, 2501, 30-42, 32-93, 37-58
to two decimal places.f Some of these have been calculated with great
accuracy.
15.3. The calculations which the above process requires are very
laborious if t is at all large. A much better method is to use the formula
D.17.5) arising from the approximate functional equation. Let us write
t = 2rru, ^ = a^ = n-lcos27T(K-u\ogn),
and
f See the references Gram F), Lindelof C), in Landau's Handbuch.

390 CALCULATIONS RELATING TO THE ZEROS Chap. XV
Then D.17.5) gives
ZBttu) =
TO
n = l
where m = [Vw], and E(u) = O(u~*). The <xn(u) can be found, for given
values of u, from a table of the function cos27nr. In the interval
0 < I < ?, h(g) decreases steadily from 0-92388 to 0-38268, and
For the purpose of calculation we require a numerical upper bound
for R{u). A rather complicated formula of this kind is obtained in
Titchmarsh A7), Theorem 2. For values of u which are not too small
it can be much simplified, and in fact it is easy to deduce that
)!< A (>
This inequality is sufficient for most purposes.
Occasionally, when ZBttu) is too small, a second term of theRiemann-
Siegel asymptotic formula has to be used.
The values of u for which the calculations are performed are the
solutions of A5.2.1), since they make <xx alternately 1 and — 1. In the
calculations described in Titchmarsh A7), I began with
u = 1-6, K = —0-04865
and went as far as
u = 62-785, K = 98-5010.
The values of u were obtained in succession, and are rather rough
approximations to the uv, so that the K's are not quite integers or
integers and a half.
It was shown in this way that the first 198 zeros of ?(s) above the real
axis all lie on the line a = \.
The calculations were carried a great deal farther by Dr. Comrie.f
Proceeding on the same lines, it was shown that the first 1,041 zeros
of ?(s) above the real axis all lie on the critical line, in the interval
0 < t < 1,468.
One interesting point which emerges from these calculations is that
Z(tv) does not always have the same sign as (—1)". A considerable
number of exceptional cases were found; but in each of these cases there
is a neighbouring point t'v such that Z(t'v) has the sign of ( — l)v, and the
succession of changes of sign of Z(t) is therefore not interrupted.
15.4. As far as they go, these calculations are all in favour of the
truth of the Riemann hypothesis. Nevertheless, it may be that they do
f See Titchmarsh A8).

15.4 CALCULATIONS RELATING TO THE ZEROS 391
not go far enough to reveal the real state of affairs. At the end of the
table constructed by Dr. Comrie there are only fifteen terms in the series
for Z(t), and this is a very small number when we are dealing with
oscillating series of this kind. Indeed there is one feature of the table
which may suggest a change in its character farther on. In the main,
the result is dominated by the first term c^, and later terms more or
less cancel out. Occasionally (e.g. at K = 435) all, or nearly all, the
numbers an have the same sign, and Z(t) has a large maximum or
minimum. As we pass from this to neighbouring values of t, the first
few <xn undergo violent changes, while the later ones vary comparatively
slowly. The term an appears when u = n2, and here
cos27r(K—u\ogn) = cos7r{u\og(u/n2)—u—§ +
= COS7TGl2 + iH--) = (-1)|
and
— (K-u\ogn) = llogu-logn-—-=- + ....
At its first appearance in the table <xn will therefore be approximately
( —l)n7i~icos^7r, and it will vary slowly for some time after its appear-
appearance.
It is conceivable that if t, and so the number of terms, were large
enough, there might be places where the smaller slowly varying terms
would combine to overpower the few quickly varying ones, and so
prevent the graph of Z(t) from crossing the zero line between successive
maxima. There are too few terms in the table already constructed to
test this possibility.
There are, of course, relations between the numbers an which destroy
any too simple argument of this kind. If the Riemann hypothesis is
true, there must be some relation, at present hidden, which prevents
the suggested possibility from ever occurring at all.
No doubt the whole matter will soon be put to the test of modern
methods of calculation. Naturally the Riemann hypothesis cannot be
proved by calculation, but, if it is false, it could be disproved by the
discovery of exceptions in this way.
NOTES FOR CHAPTER 15
15.5. A number of workers have checked the Riemann hypothesis
over increasingly large ranges. At the time of writing the most extensive
calculation is that of van de Lune and te Riele (as reported in Odlyzko
and te Riele [1 ]), who have found that the first 15 x 109 non-trivial zeros
are simple and lie on the critical line.

ORIGINAL PAPERS
[This list includes that given in my Cambridge tract; it does not include papers referred to
in Landau's Handbuch der Lehre von der Verteilung der Primzahlen, 1909.]
ABBREVIATIONS
A.M. Ada Mathematica.
C.R. Comptes rendus de VAcademic des sciences (Paris).
J.L.M.S. Journal of the London Mathematical Society.
J.M. Journal fur die reine und angewandte Mathematik.
M.A. Mathematische Annalen.
M.Z. Mathematische Zeitschrift.
P.C.P.S. Proceedings of the Cambridge Philosophical Society.
P.L.M.S. Proceedings of the London Mathematical Society.
Q.J.O. Quarterly Journal of Mathematics (Oxford Series).
Ananda-Ratt, K.
A) The infinite product for (s-l)?(s), M.Z. 20 A924), 156-64.
Abwin, A.
A) A functional equation from the theory of the Riemann ?(s)-function,
Annals of Math. B), 24 A923), 359-66.
Atkinson, F. V.
A) The mean value of the zeta-function on the critical line, Q.J.O. 10 A939)
122-8.
B) The mean value of the zeta-function on the critical line, P.L.M.S. B), 47
A941), 174-200.
C) A divisor problem, Q.J.O. 12 A941), 193-200.
D) A mean value property of the Riemann zeta-function, J.L.M.S. 23 A948),
128-35.
E) The Abel summation of certain Dirichlet series, Q.J.O. 19 A948), 59-64.
F) The Riemann zeta-function, Duke Math. J. 17 A950), 63-8.
Babini, J.
A) tjber einige Eigenschaften der Riemannschen ?(z)-Funktion, An. Soc. Ci.
Argent. 118 A934), 209-15.
Backlxjnd, R.
A) Einige numerische Rechnungen, die Nullpunkte der Riemannschen ?-
Funktionbetreffend, OfversigtFinskaVetensk. Soc. (A), 54 A911-12),No. 3.
B) Sur les zeros de la fonction ?(s) de Riemann, C.R. 158 A914), 1979-81.
C) tjber die Nullstellen der Riemannschen Zetafunktion, A.M. 41 A918),
345-75.
D) tjber die Beziehung zwischen Anwachsen und Nullstellen der Zetafunktion
Ofversigt Finska Vetensk. Sec. 61 A918-19), No. 9.
Beaupain, J.
A) Svir la fonction ?(s, w) et la fonction ?(s) de Riemann, Acad. Royale de
Belgique B), 3 A909), No. 1.
Bellman, R.
A) The Dirichlet divisor problem, Duke Math. J. 14 A947), 411-17.
B) An analog of an identity due to Wilton, Duke Math. J. 16 A949), 539-45.
C) Wigert's approximate functional equation and the Riemann zeta-function,
Duke Math. J. 16 A949), 547-52.

ORIGINAL PAPERS 393
Bohr, H.
A) En Saetning om ?-Functionen, Nyt. Tidss. for Math. (B), 21 A910), 61-6.
B) tJber das Verhalten von ?(s) in der Halbebene a > 1, Gottinger Nachrichten
A911), 409-28.
C) Sur l'existence de valeurs arbitrairement petites de la fonction ?(s) = ?(cr-{-it)
de Riemann pour a > 1, Oversigt Vidensk. Selsk. Kebenhavn A911), 201-8.
D) Sur la fonction ?(s) dans le demi-plan a > 1, C.R. 154 A912), 1078-81.
E) Uber die Funktion ?'(«)/?(«)> J-M- 141 A912), 217-34.
F) En nyt Bevis for, at den Riemann'ske Zetafunktion ?(s) = t,(a-{-it) har
uendelij mange Nulpunkten indenfor Parallel-strimlen 0 ^ a ^ 1, Nyt.
Tidss. for Math. (B) 23 A912), 81-5.
G) Om de Vaerdier, den Riemann'ske Funktion ?,(a-{-it) antager i Halvplanen
a > 1, 2. Skand. Math. Kongr. A912), 113-21.
(8) Note sur la fonction zeta de Riemann ?(s) = ^{a + it) sur la droite a = 1,
Oversigt Vidensk. Selsk. Kebenhavn A913), 3-11.
(9) Losung des absoluten Konvergenzproblems einer allgemeinen Klasse
Dirichletscher Reihen, A.M. 36 A913), 197-240.
A0) Sur la fonction ?(s) de Riemann, C.R. 158 A914), 1986-8.
A1) Zur Theorie der Riemannschen Zetafunktion im kritischen Streifen, A.M.
40 A915), 67-100.
A2) Die Riemannsche Zetafunktion, Deutsche Math. Ver. 24 A915), 1-17.
A3) tiber eine quasi-periodische Eigenschaft Dirichletscher Reihen mit An-
wendung auf die Dirichletschen L-Funktionen, M.A. 85 A922), 115-22.
A4) Uber diophantische Approximationen und ihre Anwendungen auf Dirich-
letsche Reihen, besonders auf die Riemannsche Zetafunktion, 5. Skand.
Math. Kongr. A923), 131-54.
A5) Another proof of Kronecker's theorem, P.L.M.S. B), 21 A922), 315-16.
A6) Again the Kronecker theorem, J.L.M.S. 9 A934), 5-6.
Bohb, H., and Coubant, R.
A) Neue Anwendungen der Theorie der Diophantischen Approximationen auf
die Riemannsche Zetafunktion, J.M. 144 A914), 249-74.
Bohb, H., and Jessen, B.
A), B) tiber die Werteverteilung der Riemannschen Zetafunktion, A.M. 54
A930), 1-35 and ibid. 58 A932), 1-55.
C) One more proof of Kronecker's theorem, J.L.M.S. 7 A932), 274-5.
D) Mean-value theorems for the Riemann zeta-function, Q.J.O. 5 A934), 43-7.
E) On the distribution of the values of the Riemann zeta-function, Amer. J.
Math. 58 A936), 35-44.
Bohr, H., and Landatt, E.
A) Uber das Verhalten von ?(s) und ?#(s) in der Nahe der Geraden a = 1,
Gottinger Nachrichten A910), 303-30.
B) Uber die Zetafunktion, Rend, di Palermo, 32 A911), 278-85.
C) Beitrage zur Theorie der Riemannschen Zetafunktion, M.A. 14c A913), 3—30.
D) Ein Satz uber Dirichletsche Reihen mit Anwendung auf die ^-Funktion
und die L-Funktionen, Rend, di Palermo, 37 A914), 269-72.
E) Sur les zeros de la fonction ?(s) de Riemann, C.R. 158 A914), 106-10.
F) Uber das Verhalten von l/?(s) auf der Geraden a = 1, Gottinger Nach-
Nachrichten A923), 71-80.
G) Nachtrag zu unseren Abhandlungen aus den Jahrgangen 1910 und 1923,
Gottinger Nachrichten A924), 168-72.

394 ORIGINAL PAPERS
Bohr, H., Landau, E., and Littlewood, J. E.
A) Sur la fonction ?(s) dans le voisinage de la droite a = \, Bull. Acad.
Belgique, 15 A913), 1144-75.
Bobchsenixjs, V., and Jesskn, B.
A) Mean motions and values of the Riemann zeta-function, A.M. 80 A948),
97-166.
BOTJWKAMP, C. J.
A) tjber die Riemannsche Zetafunktion fur positive, gerade Werte des Argu-
mentes, Nieuw Arch. Wisk. 19 A936), 50-8.
Bbika, M.
A) tjber eine Gestalt der Riemannschen Reihe ?(s) fiir s — gerade ganze Zahl,
Bull. Soc. Math. Grece, 14 A933), 36-8.
Bbxjn, V.
A) On the function [x], P.C.P.S. 20 A920), 299-303.
B) Deux transformations elementaires de la fonction zeta de Riemann, Revista
Ci. Lima, 41 A939), 517-25.
BUHBATJ, C.
oo
2~D log 2 "SP Vn D log p
A) Numerische Losung der Gleichung -— ^ = y — q^ wo pn die
n = 2
Reihe der Primzahlen von 3 an durchlauft, J.M. 142 A912), 51-3.
Carlson, F.
A) tiber die Nullstellen der Dirichletschen Reihen und der Riemannschen
?-Funktion, Arkiv for Mat. Astr. och Fysik, 15 A920), No. 20.
B), C) Contributions a la th^orie des series de Dirichlet, Arkiv for Mat. Astr.
och Fysik, 16 A922), No. 18, and 19 A926), No. 25.
Chowla, S. D.
A) On some identities involving zeta-functions, Journal Indian Math. Soc. 17
A928), 153-63.
COBPXJT, J. G. VAN DEB
A) Zahlentheoretische Abschatzxtngen, M.A. 84 A921), 53-79.
B) Verscharfung der Abschatzung beim Teilerproblem, M.A. 87 A922), 39-65.
C) Neue zahlentheoretische Abschatzungen, erste Mitteilung, M.A. 89 A923),
215-54.
D) Zum Teilerproblem, M.A. 98 A928), 697-716.
E) Zahlentheoretische Abschatzungen, mit Anwendung auf Gitterpunkt-
probleme, M.Z. 28 A928), 301-10.
F) Neue zahlentheoretische Abschatzungen, zweite Mitteilung, M.Z. 29 A929),
397-426.
G) tiber Weylsche Summen, Mathematica B A936-7), 1-30.
Cobpxjt, J. G. van deb, and Koksma, J. F.
A) Sur l'ordre de grandextr de la fonction ?(s) de Riemann dans la bande
critique, Annales de Toulouse C), 22 A930), 1-39.
Cbaig, C. F.
A) On the Riemann ^-Function, Bull. Amer. Math. Soc. 29 A923), 337-40.
Cbameb, H.
A) tiber die Nullstellen der Zetafunktion, M.Z. 2 A918), 237-41.
B) Studien iiber die Nullstellen der Riemannschen Zetafunktion, M.Z. 4A919),
104-30.

ORIGINAL PAPERS 395
C) Bemerkung zu der vorstehenden Arbeit des Herm E. Landau, M.Z. 6
A920), 155-7.
D) tiber das Teilerproblem von Piltz, Arkivfor Mat. Astr. och. Fysik, 16 A922),
No. 21.
E) Ein Mittelwertsatz in der Primzahltheorie, M.Z. 12 A922), 147-53.
Cram&r, H., and Landau, E.
A) tiber die Zetafunktion auf der Mittellinie des kritischen Streifens, Arkiv
for Mat. Astr. och Fysik, 15 A920), No. 28.
Crum, M. M.
A) On some. Dirichlet series, J.L.M.S. 15 A940), 10-15.
Davenport, H.
A) Note on mean-value theorems for the Riemann zeta-function, J.L.M.S. 10
A935), 136-8.
Davenport, H., and Heilbronn, H.
A), B) On the zeros of certain Dirichlet series I, II, J.L.M.S. 11 A936), 181-5
and 307-12.
Denjoy, A.
A) L'hypothese de Riemann sur la distribution des zeros de ?(s), relive a la
theorie des probability, C.R. 192 A931), 656-8.
Deuring, M.
A) Imaginare quadratische Zahlkorper mit der Klassenzahl 1, M.Z. 37 A933),
405-15.
B) On Epstein's zeta-function, Annals of Math. B) 38 A937), 584-93.
ESTERMANN, T.
A) On certain functions represented by Dirichlet series, P.L.M.S. B), 27
A928), 435-48.
B) On a problem of analytic continuation, P.L.M.S. 27 A928), 471-82.
C) A proof of Kronecker's theorem by induction, J.L.M.S. 8 A933), 18-20.
Favard, J.
A) Sur la repartition des points ou une fonction presque periodique prend une
valeur donnee, C.R. 194 A932), 1714-16.
Fejer, L.
A) Nombre de changements de signe d'une fonction dans un intervalle et ses
moments, C.R. 158 A914), 1328-31.
Fekete, M.
A) Sur une limite inferieure des changements de signe d'une fonction dans un
intervalle, C.R. 158 A914), 1256-8.
B) The zeros of Riemann's zeta-function on the critical line, J.L.M.S. 1 A926),
15-19.
Flett, T. M.
00
A) On the function V -sin-, J.L.M.S. 25 A950), 5-19.
A n n
n = l
B) On a coefficient problem of Littlewood and some trigonometrical sums,
Q.J.O. B) 2 A951), 26-52.
Franel, J.
A) Les suites de Farey et le probleme des nombres premiers, Gottinger Nach-
richten A924), 198-201.

396 ORIGINAL PAPERS
Gabriel, R. M.
A) Some results concerning the integrals of moduli of regular functions along
certain curves, J.L.M.S. 2 A927), 112-17.
Gram, J. P.
A) Tafeln fur die Riemannsche Zetafunktion, Skriften Kebenharn (8), 9 A925),
311-25.
Gronwall, T. H.
A) Sur la fonction ?(s) de Riemann au voisinage de cr = 1, Rend, di Palermo,
35 A913), 95-102.
B) tiber das Verhalten der Riemannschen Zeta-funktion auf der Geraden
a = 1, Arch, der Math. u. Phys. C) 21 A913), 231-8.
Grossman, J.
A) tiber die Nullstellen der Riemannschen Zeta-funktion und der Dirich-
letschen .L-Funktionen, Dissertation, Gottingen A913).
GuiNAND, A. P.
A) A formula for ?(s) in the critical strip, J.L.M.S. 14 A939), 97-100.
B) Some Fourier transforms in prime-number theory, Q.J.O. 18 A947), 53-
64.
C) Some formulae for the Riemann zeta-function, J.L.M.S. 22 A947), 14-18.
D) Fourier reciprocities and the Riemann zeta-function, P.L.M.S. 51 A949),
401-14.
Hadamard, J.
A) Une application d'une formule integrate relative aux series de Dirichlet,
Bull. Soc. Math, de France, 56, ii A927), 43-4.
Hamburger, H.
A), B), C) tiber die Riemannsche Funktionalgleichung der ?-Funktion, M.Z.
10 A921), 240-54; 11 A922), 224-45; 13 A922), 283-311.
D) tiber einige Beziehungen, die mit der Funktionalgleichung der Riemann-
Riemannschen ?-Funktion Equivalent sind, M.A. 85 A922), 129-40.
Hardy, G. H.
A) Sur les zeros de la fonction ?(s) de Riemann, C.R. 158 A914), 1012-14.
B) On Dirichlet's divisor problem, P.L.M.S. B), 15 A915), 1-25.
C) On the average order of the arithmetical functions P(n) and A(n), P.L.M.S.
B), 15 A915), 192-213.
D) On some definite integrals considered by Mellin, Messenger of Math. 49
A919), 85-91.
E) Ramanujan's trigonometrical function cq{n), P.C.P.S. 20 A920), 263-71.
F) On the integration of Fourier series, Messenger of Math. 51 A922), 186-92.
G) A new proof of the functional equation for the zeta-function, Mat. Tids-
skrift, B A922), 71-3.
(8) Note on a theorem of Mertens, J.L.M.S. 2 A926), 70-2.
Hardy, G. H., Ingham, A. E., and P6lya, G.
A) Theorems concerning mean values of analytic functions, Proc. Royal Soc.
(A), 113 A936), 542-69.
Hardy, G. H., and Littlewood, J. E.
A) Some problems of Diophantine approximation, Internal. Congress of Math.,
Cambridge A912), 1, 223-9.
B) Contributions to the theory of the Riemann zeta-function and the theory
of the distribution of primes, A.M. 41 A918), 119-96.

ORIGINAL PAPERS 397
C) The zeros of Riemann's zeta-function on the critical line, M.Z. 10 A921),
283-317.
D) The approximate functional equation in the theory of the zeta-function,
with applications to the divisor problems of Dirichlet and Piltz, P.L.M.S.
B), 21 A922), 39-74.
E) On Lindelof's hypothesis concerning the Riemann zeta-function, Proc.
Royal Soc. (A), 103 A923), 403-12.
F) The approximate functional equations for ?(s) and ?2(s), P.L.M.S. B), 29
A929), 81-97.
Hartman, P.
A) Mean motions and almost periodic functions, Trans. Amer. Math. Soc. 46
A939), 66-81.
Haselgrove, C. B.
A) A connexion between the zeros and the mean values of ?(s), J.L.M.S. 24
A949), 215-22.
Hasse, H.
A) Beweis des Analogons der Riemannschen Vermutung fur die Artinschen
und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen
Fallen, Gottinger Nachrichten, 42 A933), 253-62.
Haviland, E. K.
A) On the asymptotic behavior of the Riemann ?-function, Amer. J. Math. 67
A945), 411-16.
Hecke, E.
A) tiber die Losungen der Riemannschen Funktionalgleichung, M.Z. 16
A923), 301-7.
B) tiber die Bestimmung Dirichletscher Reihen durch ihre Funktional-
Funktionalgleichung, M.A. 112 A936), 664-99.
C) Herleitung des Euler-Produktes der Zetafunktion und einiger L -Reihen
aus ihrer Funktionalgleichung, M.A. 119 A944), 266-87.
Heilbronn, H.
A) tiber den Primzahlsatz von Herrn Hoheisel, M.Z. 36 A933), 394-423.
Hole, E.
A) A problem in ' Factorisatio Numerorum', Ada Arith. 2 A936), 134-44.
Hoheisel, G.
A) Normalfolgen und Zetafunktion, Jahresber. Schles. Gesell. 100 A927), 1-7.
B) Eine Illustration zur Riemannschen Vermutung, M.A. 99 A928), 150—61.
C) Uber das Verhalten des reziproken Wertes der Riemannschen Zeta-
Funktion, Sitzungsber. Preuss. Akad. Wiss. A929), 219-23.
D) Nullstellenanzahl und Mittelwerte der Zetafunktion, Sitzungsber. Preuss.
Akad. Wiss. A930), 72-82.
E) Primzahlprobleme in der Analysis, Sitzungsber. Prettss. Akad. Wiss. A930),
580-8.
Holder, O.
A) tiber gewisse der Mobiusschen Funktion fj.(n) verwandte zahlentheoretische
Funktionen, der Dirichletschen Multiplikation und eine Verallgemeine-
rung der Urnkehrungsformeln, Ber. Verh. sdchs. Akad. Leipzig, 85 A933),
3-28.
Hua, L. K.
A) An improvement of Vinogradov's mean-value theorem and several applica-
applications, Q.J.O. 20 A949), 48-61.

398 ORIGINAL PAPERS
HlJTCHINSON, J. I.
A) On the roots of the Riemann zeta-function, Trans. Amer. Math. Soc. 27
A925), 49-60.
LSTGHAM, A. E.
A) Mean-value theorems in the theory of the Riemann zeta-function, P.L.M.S.
B), 27 A926), 273-300.
B) Some asymptotic formulae in the theory of numbers, J.L.M.S. 2 A927),
202-8.
C) Notes on Riemann's ^-function and Dirichlet's .L-functions, J.L.M.S. 5
A930), 107-12.
D) Mean-value theorems and the Riemann zeta-function, Q.J.O.4A933), 278-90.
E) On the difference between consecutive primes, Q.J.O. 8 A937), 255-66.
F) On the estimation of N(cr, T), Q.J.O. 11 A940), 291-2.
G) On two conjectures in the theory of numbers, Amer. J. Math. 64 A942),
313-19.
JabnIk, V., and Landau, E.
A) Untersuchungen iiber einen van der Corputschen Satz, M.Z. 39 A935),
745-67.
Jessen, B.
A) Eine Integrationstheorie fur Funktionen unendlich vieler Veranderlichen,
mit Anwendung auf das Werteverteilungsproblem fur fastperiodische
Funktionen, insbesondere fur die Riemannsche Zetafunktion, Mat.
Tidsskrift, B A932), 59-65.
B) Mouvement moyen et distribution des valeurs des fonctions presque-
periodiques, 10. Skand. Math. Kongr. A946), 301-12.
Jessen, B., and Wintner, A.
A) Distribution functions and the Riemann zeta-function, Trans. Amer. Math.
Soc. 38 A935), 48-88.
Kac, M., and Steinhaus, H.
A) Sur les fonctions independantes (IV), Studia Math. 7 A938), 1-15.
Kampen, E. R. van
A) On the addition of convex curves and the densities of certain infinite
convolutions, Amer. J. Math. 59 A937), 679-95.
Kampen, E. R. van, and Wintner, A.
A) Convolutions of distributions on convex curves and the Riemann zeta-
function, Amer. J. Math. 59 A937), 175-204.
Kershner, R.
A) On the values of the Riemann ^-function on fixed lines cr > 1, Amer. J.
Math. 59 A937), 167-74.
Kershner, R., and Wintner, A.
A) On the boundary of the range of values of ?(s), Amer. J. Math. 58 A936),
421-5.
B) On the asymptotic distribution of ?'/?(s) ^ ^ne critical strip, Amer. J.
Math. 59 A937), 673-8.
KlENAST, A.
A) tiber die Dirichletschen Reihen fur ?p(s), Lp(s), Comment. Math. Helv. 8
A936), 359-70.
Kloosterman, H. D.
A) Een integraal voor de ?-functie van Riemann, Ghristiaan Huygens Math.
Tijdschrift, 2 A922), 172-7.

ORIGINAL PAPERS 399
Kluyver, J. C.
A) On certain series of Mr. Hardy, Quart. J. of Math. 50 A924), 185-92.
Kober, H.
A) Transformationen einer bestimmten Besselschen Reihe sowie von Potenzen
der Riemannschen ?-Funktion und von verwandten Funktionen, J.M.
173 A935), 65-78.
B) Eine der Riemannschen verwandte Funktionalgleichung, M.Z. 39 A935),
630-3.
C) Funktionen, die den Potenzen der Riemannschen Zetafunktion verwandt
sind, und Potenzreihen, die iiber den Einheitskreis nicht fortsetzbar sind,
J.M. 174 A936), 206-25.
D) Eine Mittelwertformel der Riemannschen Zetafunktion, Compositio Math.
3 A936), 174-89.
Koch, H. von
A) Contribution a la theorie des nombres premiers, A.M. 33 A910), 293-320.
Kosliakov, N.
A) Some integral representations of the square of Riemann's function S(?),
C.R. Acad. Sci. U.R.S.S. 2 A934), 401-4.
B) Integral for the square of Riemann's function, C.R. Acad. Sci. U.R.S.S.
n.s. 2 A936), 87-90.
C) Some formulae for the functions ?(s) and ?r(s), C.R. Acad. Sci. U.R.S.S.
B), 25 A939), 567-9.
Rramaschke, L.
A) Nullstellen der Zetafunktion, Deutsche Math. 2 A937), 107-10.
Kusmin, R.
A) Sur les zero de la fonction ?(s) de Riemann, C.R. Acad. Sci. U.R.S.S. 2
A934), 398-400.
Landau, E.
A) Zur Theorie der Riemannschen Zetafunktion, Vierteljahrsschr. Naturf. Ges.
Zurich. 56 A911), 125-48.
B) tiber die Nullstellen der Zetafunktion, M.A. 71 A911), 548-64.
C) Ein Satz iiber die ?-Funktion, Nyt. Tidss. 22 (B), A911), 1-7.
D) tjber einige Summen, die von den Nullstellen der Riemannschen Zeta-
Zetafunktion abhangen, A.M. 35 A912), 271-94.
E) tjber die Anzahl der Gitterpunkte in gewissen Bereichen, Gottinger Nach-
richten A912), 687-771.
F) Geloste and ungeloste Probleme aus der Theorie der Primzahlverteilung
und der Riemannschen Zetafunktion, Jahresber. der Deutschen Math. Ver.
21 A912), 208-28.
G) Geloste and ungeloste Probleme aus der Theorie der Primzahlverteilung
und der Riemannschen Zetafunktion, Proc. 5 Internat. Math. Congr.
A913), 1, 93-108.
(8) tiber die Hardysche Entdeckung unendlich vieler Nullstellen der Zeta-
Zetafunktion mit reellem Teil J, M.A. 76 A915), 212-43.
(9) tjber die Wigertsche asymptotische Funktionalgleichung fur die Lam-
bertsche Reihe, Arch. d. Math. u. Phys. C), 27 A916), 144-6.
A0) Neuer Beweis eines Satzes von Herrn Valiron, Jahresber. der Deutschen
Math. Ver. 29 A920), 239.
A1) tiber die Nullstellen der Zetafunktion, M.Z. 6 A920), 151-4.

400 ORIGINAL PAPERS
A2) tiber die Nullstellen der Dirichletschen Reihen und der Riema.nnschen
?-Funktion, Arkiv for Mat. Astr. och Fysik, 16 A921), No. 7.
A3) tiber die Mobiussche Funktion, Rend, di Palermo, 48 A924), 277-80.
A4) tiber die Wurzeln der Zetafunktion, M.Z. 20 A924), 98-104.
A5) tiber die ?-Funktion und die ?-Funktionen, M.Z. 20 A924), 105-25.
A6) Bemerkung zu der vorstehenden Arbeit von Herrn Franel, Gbttinger
Nachrichten A924), 202-6.
A7) tiber die Riemannsche Zetafunktion in der Nahe von a == 1, Rend, di
Palermo, 50 A926,), 423-7.
A8) Uber die Zetafunktion und die Hadamardsche Theorie der ganzen Funkti-
onen, M.Z. 26 A927), 170-5.
A9) tiber das Konvergenzgebiet einer mit der Riemannschen Zetafunktion
zusammenhangenden Reihe, M.A. 97 A927), 251-90.
B0) Bemerkung zu einer Arbeit von Hm. Hoheisel uber die Zetafunktion,
Sitzungsber. Prcuss. Akad. Wiss. A929), 271-5.
B1) tiber die Fareyreihe und die Riemannsche Vermutung, Gb'ttinger Nach-
Nachrichten A932), 347-52.
B2) tiber den Wertevorrat von ?(s) in der Halbebene cr > 1, Gb'ttinger Nach-
Nachrichten A933), 81-91.
Landau, E., and Walfisz, A.
A) tiber die Nichtfortsetzbarkeit einiger durch Dirichletsche Reihen definierter
Funktionen, Rend, di Palermo 44 A919), 82-6.
Lerch, M.
A) tiber die Bestimmung der Koeffizienten in der Potenzreihe fur die Funktion
?(«), Gasopis, 43 A914), 511-22.
Lettenmeyer, F.
A) Neuer Beweis des allgemeinen Kroneckerschen Approximationssatzes,
P.L.M.S. B) 21 A923), 306-14.
Levinson, N.
A) On Hardy's theorem on the zeros of the zeta function, J. Math. Phys.
Mass. Inst. Tech. 19 A940), 159-60.
Little wood, J. E.
A) Quelques consequences de l'hypothese que la fonction ?(s) de Riemann n'a
pas de zeros dans le demi-plan R(s) > \, CR. 154 A912), 263-6.
B) Researches in the theory of the Riemann ^-function, P.L.M.S. B) 20 A922),
Records xxii-xxviii.
C) Two notes on the Riemann zeta-function, P.C.P.S. 22 A924), 234-42.
D) On the zeros of the Riemann zeta-function, P.C.P.S. 22 A924), 295-318.
E) On the Riemann zeta-function, P.L.M.S. B), 24 A925), 175-201.
F) On the function l/?(l+ri), P.L.M.S. B), 27 A928), 349-57.
Maier, W.
A) Gitterfunktionen der Zahlebene, M.A. 113 A936), 363-79.
Malurkar, S. L.
A) On the application of Heir Mellin's integrals to some series, Journal Indian
Math. Soc. 16 A925), 130-8.
Mattson, R.
A) Eine neue Darstellung der Riemann'schen Zetafunktion, Arkiv for Mat.
Astr. och Fysik, 19 A926), No. 26.

ORIGINAL PAPERS 401
Mellin, H.
A) tiber die Nullstellen der Zetafunktion, Annales Acad. Scientiarium Fen-
nicae (A), 10 A917), No. 11.
Meulenbeld, B.
A) Een approximative Functionaalbetrekking van de Zetafunctie van
Riemann, Dissertation, Groningen, 1936.
Mikolas, M.
A) Sur l'hypothese de Riemann, C.R. 228 A949), 633-6.
Min, S. H.
A) On the order of t,(% + it), Trans. Amer. Math. Soc. 65 A949), 448-72.
MlYATAKE, O.
A) On Riemann's ^-function, Tohoku Math. Journal, 46 A939), 160-72.
Mordell, L. J.
A) Some applications of Fourier series in the analytic theory of numbers,
P.C.P.S. 34 A928), 585-96.
B) Poisson's summation formula and the Riemann zeta-function, J.L.M.S. 4
A929), 285-91.
C) On the Riemann hypothesis and imaginary quadratic fields with a given
class number, J.L.M.S. 9 A934), 289-98.
Muntz, C. H.
A) Beziehungen der Riemannschen ?-Funktion zu willkiirlichen reellen
Funktionen, Mat. Tidsskrift, B A922), 39-47.
Mutatker, V. L.
A) On some formulae in the theory of the zeta-function, Journal Indian Math.
Soc. 19 A932), 220-4.
Nevanlinna, F. and R.
A), B) tiber die Nullstellen der Riemannschen Zetafunktion, M.Z. 20 A924),
253-63, and 23 A925), 159-60.
OSTROWSKI, A.
A) Notiz iiber den Wertevorrat der Riemannschen ?-Funktion am Rande des
kritischen Streifens, Jahresbericht Deutsch. Math. Verein. 43 A933),
58-64.
Paley, R. E. A. C, and Wiener, N.
A) Notes on the theory and application of Fourier transforms V, Trans. Amer.
Math. Soc. 35 A933), 768-81.
Phillips, Eric
A) The zeta-function of Riemann; further developments of van der Corput's
method, Q.J.O. 4 A933), 209-25.
B) A note on the zeros of ?(s), Q.J.O. 6 A935), 137-45.
Pol, B. van der,
A) An electro-mechanical investigation of the Riemann zeta-function in the
critical strip, Bull. Amer. Math. Soc. 53 A947), 970-81.
Polya, G.
A) Bemerkung uber die Integraldarstellung der Riemannschen ?-Funktion,
A.M. 48 A926), 305-17.
B) On the zeros of certain trigonometric integrals, J-L.M.S. 1 A926), 98-9.
C) Uber die algebraisch-funktiontheoretischen Untersuchungen von J. L. W. V.
Jensen, Kgl. Danslce Videnskabernes Selskab. 7 A927), No. 17.
D) Uber trigonometrische Integrate mit nur reellen Nullstellen, J.M. 15S
A927), 6-18.

402 ORIGINAL PAPERS
Popov, A. I.
A) Several series containing primes and roots of ?(*), C.R. Acad. Sci. U.R.S.S
n.s. 41 A943), 362-3.
Potter, H. S. A., and Titchmarsh, E. C.
A) The zeros of Epstein's zeta-functions, P.L.M.S. B) 39 A935), 372-84.
Rademacher, H.
A) Ein neuer Beweis fur die Funktionalgleichung der ?-Funktion. M.Z. 31
A930), 39-44.
PvAMANTJJAN, S.
A) New expressions for Riemann's functions ?(s) and E(t), Quart. J. of Math.
46 A915), 253-61.
B) Some formulae in the analytic theory of numbers, Messenger of Math. 45
A915), 81-4.
C) On certain trigonometrical sums and their applications in the theory of
numbers, Trans. Camb. Phil. Soc. 22 A918), 259-76.
Ramaswami, V.
A) Notes on Riemann's ^-function, J.L.M.S. 9 A934), 165-9.
Riesz, M.
A) Sur l'hypothese de Riemann, A.M. 40 A916), 185-90.
Schnee, W.
A) Die Funktionalgleichung der -Zetafunktion und der Dirichletschen Reihen
mit periodischen Koeffizienten, M.Z. 31 A930), 378-90.
Selberg, A.
A) On the zeros of Riemann's zeta-function on the critical line, Arch, for Math.
og Naturv. B, 45 A942), 101-14.
B) On the zeros of Riemann's zeta-function, Skr. Norske Vid. Akad. Oslo
A942), no. 10.
C) On the normal density of primes in small intervals, and the difference
between consecutive primes, Arch, for Math, og Naturv. B, 47 A943), No. 6.
D) On the remainder in the formula for N(T), the number of zeros of ?(s) in
the strip 0 < t < T, Avhandlinger Norske Vid. Akad. Oslo A944), no. 1.
E) Contributions to the theory of the Riemann zeta-function, Arch, for Math.
og Naturv. B, 48 A946), no. 5.
F) The zeta-function and the Riemann hypothesis, 10. Skand. Math. Kongr.
A946), 187-200.
G) An elementary proof of the prime-number theorem, Ann. of Math. B) 50
A949), 305-13.
Selberg, S.
A) Bemerkung zu einer Arbeit von Viggo Brun iiber die Riemannsche Zeta-
Zetafunktion, Norske Vid. Selsk. Forh. 13 A940), 17-19.
SlEGEL, C. L.
A) Bemerkung zu einem Satz von Hamburger iiber die Funktionalgleichung
der Riemannschen Zetafunktion, M.A. 86 A922), 276-9.
B) tiber Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und
Studien zur Geschichte der Math. Astr. und Physik, Abt. B: Studien,
2 A932), 45-80.
C) Contributions to the theory of the Dirichlet L -series and the Epstein
Zeta-functions, Annals of Math. 44 A943), 143-72.
Speiser, A.
A) Geometrisches zur Riemannschen Zetafunktion, M.A. 110 A934), 514-21.

ORIGINAL PAPERS 403
Steen, S. W. P.
A) A linear transformation connected with the Riemann zeta-function,
P.L.M.S. B), 41 A936), 151-75.
Stekneck, R. von
A) Neue empirische Daten uber die zahlentheoretische Funktion a{n), Internat.
Congress of Math., Cambridge A912), 1, 341-3.
SzAsz, O.
A) Introduction to the theory of divergent series, Cincinnati, 1944 (Math.
Rev. 6, 45).
Taylor, P. R.
A) On the Riemann zeta-function, Q.J.O. 16 A945), 1-21.
TCHUDAKOJFF, N. G.
A) Sur les zeros de la fonction ?(«), C.R. 202 A936), 191-3.
B) On zeros of the function ?(s), C.R. Acad. Sci. U.R.S.S. 1 (x) A936), 201-4.
C) On zeros of Dirichlet's ^-functions, Mat. Sbornik A) 43 A936), 591-602.
D) On Weyl's sums, Mat. Sbornik B) 44 A937), 17-35.
E) On the functions ?(s) and tt{x), C.R. Acad. Sci. U.R.S.S. 21 A938), 421-2.
Thirtjvenkatachabya, V.
A) On some properties of the zeta-function, Journal Indian Math. Soc. 19
A931), 92-6.
TlTCHMABSH, E. C.
A) The mean-value of the zeta-function on the critical line, P.L.M.S. B) 27
A928), 137-50.
B) On the remainder in the formula for N(T), the number of zeros of ?(s) in
the strip 0 < t < T, P.L.M.S. B), 27 A928), 449-58.
C) A consequence of the Riemann hypothesis, J.L.M.S. 2 A927), 247-54.
D) On an inequality satisfied by the zeta-function of Riemann, P.L.M.S. B),
28 A928), 70-80.
E) On the zeros of the Riemann zeta-function, P.L.M.S. B) 30 A929), 319-21.
F) Mean value theorems in the theory of the Riemann zeta-function, Messenger
of Math. 58 A929), 125-9.
G) The zeros of Dirichlet's ^-functions, P.L.M.S. B) 32 A931), 488-500.
(8)-A2) On van der Corput's method and the zeta-function of Riemann,
Q.J.O. 2 A931), 161-73; 2 A931), 313-20; 3 A932), 133-41; 5 A934),
98-105; 5 A934), 195-210.
A3) On the Riemann zeta-function, P.C.P.S. 28 A932), 273-4.
A4) On the function l/?(l+&), Q.J.O. 4 A933), 64-70.
A5) On Epstein's zeta-function, P.L.M.S. B) 36 A934), 485-500.
A6) The lattice-points in a circle P.L.M.S. B) 38 A935), 96-115.
A7), A8) The zeros of the Riemann zeta-function, Proc. Royal Soc. (A), 151
A935), 234-55, and ibid. 157 A936), 261-3.
A9) The mean value of |?(?+*OI4> Q.J.O. 8 A937), 107-12.
B0) On ?(«) and tt(x), Q.J.O. 9 A938), 97-108.
B1) The approximate functional equation for ?2(s), Q.J.O. 9 A938), 109-14.
B2) On divisor problems, Q.J.O. 9 A938), 216-20.
B3) A convexity theorem, J.L.M.S. 13 A938), 196-7.
B4) On the order of ?(?+&), Q.J.O. 13 A942), 11-17.
B5) Some properties of the Riemann zeta-function, Q.J.O. 14 A943), 16-26.
B6) On the zeros of the Riemann zeta function, Q.J.O. 18 A947), 4-16.

404 ORIGINAL PAPERS
TORELLI, G.
A) Studio sulla funzione ?(s) di Riemann, Napoli Rend. C) 19 A913), 212-16.
Tsuji, M.
A) On the zeros of the Riemann zeta-function, Proc. Imp. Acad. Tokyo, 18
A942), 631-4.
TtjrAn, P.
A) tJber die Verteilung der Primzahlen I, Ada Szeged, 10 A941), 81-104.
B) On Riemann's hypothesis, Bull. Acad. Sci. U.R.S.S. 11 A947), 197-262.
C) On some approximative Dirichlet-polynomials in the theory of the zeta-
function of Riemann, Danske Vidensk. Selskab, 24 A948), Nr. 17.
Turing, A. M.
A) A method for the calculation of the zeta-function, P.L.M.S. B), 48 A943),
180-97.
Utzinger, A. A.
A) Die reellen Ziige der Riemannschen Zetafunktion, Dissertation, Zurich,
1934 (see Zentralblatt fur Math. 10, 163).
Vaxiron, G.
A) Sur les fonctions entieres d'ordre nul et d'ordre fini, Annales de Toulouse
C), 5 A914), 117-257.
Vallee Potjssin, C. de la
A), B) Sur les zeros de ?(s) de Riemann, C.R. 163 A916), 418-21 and 471-3.
VlNOGRADOV, I. M.
A) On Weyl's sums, Mat. Sbornik, 42 A935), 521-30.
B) A new method of resolving of certain general questions of the theory of
numbers, Mat. Sbornik A), 43 A936), 9-19.
C) A new method of estimation of trigonometrical sums, Mat. Sbornik A), 43
A936), 175-188.
D) The method of trigonometrical sums in the theory of numbers, Trav. Inst.
Math. Stekloff 23 A947).
Voronoi, G.
A) Sur un probleme du calcul des fonctions asymptotiques, J.M. 126 A903),
241-82.
B) Sur une fonction transcendante et ses applications a la sommation de quelques
series, Annales de Vficole Normale C) 21 A904), 207-68 and 459-534.
Wajlfisz, A.
A) Zur Abschatzung von t,{\ + it), Gottinger Nachrichten A924), 155-8.
B) tiber Gitterpunkte in mehrdimensionalen Ellipsoiden VIII, Travaux de
VInstitut Math, de Tbilissi, 5 A938), 181-96.
Wajlther, A.
A) IJber die Extrema der Riemannschen Zetafunktion bei reellem Argument,
Jahresbericht Deutsch. Math. Verein. 34 A925), 171-7.
B) Anschauliches zur Riemannschen Zetafunktion, A.M. 48 A926), 393-400.
Wang, F. T.
A) A remark on the mean-value theorem of Riemann's zeta-function, Science
Reports Tohoku Imperial Univ. A) 25 A936), 381-91.
B) On the mean-value theorem of Riemann's zeta-function, Science Reports
Tohoku Imperial Univ. A) 25 A936), 392-414.
C) A note on zeros of Riemann zeta-function, Proc. Imp. Acad. Tokyo, 12,
A937), 305-6.
D) A formula on Riemann zeta-function, Ann. of Math. B) 46 A945), 88-92.

ORIGINAL PAPERS 405
E) A note on the Riemann zeta-function, Bull. Amer. Math. Soc. 52 A946),
319-21.
F) A mean-value theorem of the Riemann zeta function, Q.J.O. 18 A947), 1-3.
Watson, G. N.
A) Some properties of the extended zeta-function, P.L.M.S. B) 12 A913),
288-96.
Wennbebg, S.
A) Zur Theorie der Dirichletschen Reihen, Dissertation, Upsala, 1920.
Weyl, H.
A) Uber die Gleichverteilung von Zahlen mod. Eins, M.A. 77 A916), 313-52.
B) Zur Abschatzung von ?(l+«i), M.Z. 10 A921), 88-101.
Whittakeb, J. M.
A) An inequality for the Riemarm zeta-function, P.?Jkf.?. B), 41 A936), 544-52.
B) Amean-value theorem for analytic functions, P.L.M.S. B), 42 A936), 186-95.
WlGEBT, S.
A) Sur la serie de Lambert et son application a la theorie des nombres, A.M.
41 A916), 197-218.
B) Sur la theorie de la fonction ?(s) de Riemann, Arkiv for Mat. Astr. och
Fysilc A919), No. 12.
C) OnaproblemconcerningtheRiemann^-function,P.G.P.S.21 A921), 17-21.
Wilson, B. M.
A) Proofs of some formulae enunciated by Ramanujan, P.L.M.S. B) 21
A922), 235-55.
Wilton, J. R.
A) Note on the zeros of Riemann's ^-function, Messenger of Math. 45 A915),
180-3.
B) A proof of Burnside's formula for log F(a;+1) and certain allied properties
of Riemann's ^-function, Messenger of Math. 52 A922), 90-3.
C) A note on the coefficients in the expansion of ?(s, x) in powers of s— 1,
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