a. karatsuba - the riemann zeta-function.pdf

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fj , ~

I The Riemann eta-Function I

by

A. A. Karatsuba S. M. Voronin

Translated from the Russian

by Neal Koblitz

Walter de Gruyter . Berlin' New York 1992

Authors A. A. Karatsuba S. M. Voronin Department of Analytic Number Theory Steklov Mathematical Institute Vavilova 42 117966 Moscow GSP-1, Russia

Translator Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98 195 USA

1991 Mathematics Subject Classification: 11-02; 11M06,11M26, llN05, 11Ll5

@ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data

Karatsuba, A. A. (Anatolii Alekseevich) 1937-The Riemann zeta-function / by A. A. Karatsuba, S. M.

Voronin ; translated from the Russian by Neal Koblitz. p. cm. - (De Gruyter expositions in mathematics.

ISSN 0938-6572 ; 5) Includes bibliographical references and index. ISBN 3-11-013170-6 (cloth; acid-free) 1. Functions, Zeta. I. Voronin, Sergei MikhaHovich.

II. Title. III. Series. QA246.V73 1992 512'.73 -dc20 92-22817

Die Deutsche Bibliothek - Cataloging-in-Publication Data

Karacuba, Anatolij A.: The Riemann zeta-function / by A. A. Karatsuba Voronin. Trans!. from the Russ. by Neal Koblitz. New York: de Gruyter, 1992

(De Gruyter expositions in mathematics ; 5) ISBN 3-11-013170-6

NE: Voronin, Sergej M.:; GT

CIP

; S.M. Berlin;

Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, .including those of translation into foreign languages. No part of this book may b~ reproduce~ m any form or by any means, electronic or mechanical, including photocopy, recordmg, or any mformation storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Disk Conversion: D. L. Lewis, Berlin. Printing: Gerike GmbH, Berlin. Binding: Liideritz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

,. ........ --.................. . :f

'I ! ! I Preface

This monograph is devoted to a systematic exposition of the theory of the Riemann zeta-function. This type of project is not new. One need only recall Titchmarsh's The Theory of the Riemann Zeta-Function, first published in 1951 and then reissued by Oxford University Press in 1986. Titchmarsh's book has not lost its special importance, as a veritable encyclopedia of the zeta-function. At the same time, there have been certain areas where the theory of the Riemann zeta-function has made significant progress in recent years. In addition, certain aspects of the theory have been extended to a broader class of functions.

It is not the purpose of the authors to cover all of the areas of research where progress has been made. A natural criterion for selection was as much as possible to avoid duplication and rewriting of material that is already available. Our emphasis is on results which have not yet appeared in monograph form. Nevertheless, we hope that this book has a broad enough sweep so as to convey an impression of the current state of research on the Riemann zeta-function.

We are pleased to express our gratitude to N. 1. Voronina for her invaluable assistance in preparing the manuscript. We are thankful to Prof. N. Koblitz for his excellent translation of our book into English.

Moscow, April 1992 A.A. Karatsuba, S.M. Voronin

Notation

As a rule, standard mathematical notation will be used without comment. C, CI, C2, .. , C, C I, ... denote positive constants, which may have different val-

ues in different places. E, EI, ... denote arbitrarily small positive numbers. p, Po, PI, ... denote prime numbers. n, m, I, r denote either integers or natural numbers, depending on the context. log x = In x denotes the natural logarithm of x. Given a real number a, we set Iiall = min({a}, 1 - {a}), where {a} = a - [a]

is the fractional part of a. The notation A B means that there exists C > 0 such that I A I :s C B . The notation A k,N,y B means that A B holds for fixed k, N, y. For real and positive functions f and cp the notation f ~ cp means that there

exist positive CI and C2 such that cd :s cp :s c2f. In any discussion of t; - and L-functions we set S = (J + it. If s has subscripts,

then (J and t will have the same subscripts. The notation (ai, ... , an) == (b l , ... , bn) (mod 1) means that aj -b j is a rational

integer for all j = 1, ... ,n. Theorems, lemmas, corollaries and formulas are referred to by chapter, section

and subsection. If the reference is within the same chapter (or section), then the corresponding number is deleted. Roman numerals are used for chapters, and Ara-bic numerals for sections and subsections. Some preliminary results are gathered together in the Appendix, which is denoted Chapter A when these results are cited.

Introduction

The method of generating functions goes back more than two centuries to the work of Euler. Its early applications were connected with problems in number theory and combinatorial analysis. The range of applications of generating functions later broadened to include algebra, topology, and - in a very significant way -probability theory. But it is in number theory, the first field to which the method was applied, that one finds some of its most brilliant achievements.

The point of departure for the method of generating functions is the construction of functions which correspond to the objects under consideration. This is done in such a way that the properties of these objects are reflected in relations between the functions. Since one can apply the full arsenal of mathematical analysis to the functions, this often leads to valuable results in the study of the original problem.

In this book the basic object of study is the Riemann zeta-function, although to some extent we touch upon the properties of Dirichlet and Hecke L-functions as well. Many properties of the integers are reflected in the analytic properties of the zeta-function. For example, the Euler product representation for the zeta-function is a reflection of the unique factorization of integers into primes. Other connections between properties of the integers and analytic properties of the zeta-function are not so immediately obvious, although in many cases a careful analysis makes the relationship clear.

We now give a brief summary of the contents of the book. Chapter I contains an exposition of the basic facts of the theory of the Riemann

t; -function and Dirichlet L-series, regarded as functions of a complex variable. The presentation is fairly standard.

In Chapter II the Riemann zeta-function is regarded as the generating function of a series of arithmetic functions. From this point of view we prove the most commonly used identities. We pay special attention to the connection of the t;-function with the distribution of prime numbers, i.e., with the function n(x) (the number of primes not exceeding x). In particular, we derive asymptotic laws for the distribution of primes among the natural numbers and in arithmetic progressions.

In Chapter III we develop the machinery that is used to obtain the so-called approximate functional equations. These equations make it possible to approximate the Riemann zeta-function in the critical strip 0

viii Introduction

The classical method of Vinogradov for estimating Weyl sums is presented in Chapter IV. As a corollary we obtain a result on zero-free regions of the t -function. In 4 Vinogradov's method is used to obtain an asymptotic formula for the number of integer points under a hyperboloid. The remainder term in this formula is the best that is currently known.

In Chapter V we study various density theorems and their corollaries. It should be noted that even Ingham's theorem is given without proof - instead, a weaker result is proved in 2. The reason for this is that density theorems occupy a large place in the modern literature; while for our subsequent purposes the result proved in 2 is completely sufficient. The reader who is particularly interested in Ingham type density theorems may consult the references cited in the remarks.

The results in 5 of Chapter V on the distlibution of zeros in a neighborhood of the line Re s = 1/2 are in monograph form apparently for the first time.

In Chapter VI we study the zeros of the Riemann zeta-function on the line Re s = 1/2. We describe the current state of know ledge in this area, which was the subject of various papers by Hardy and Selberg.

In Chapter VII we extend the investigations of Harald Bohr on the distribution of values of the t -function. We give some generalizations of results on the Riemann t -function to Dirichlet and Hecke L-functions.

Chapter VIII contains an exposition of classical results and also some new results making effective the theorems of Chapter VII.

Preface Notation Introduction

Table of Contents

Chapter I

V

VI

vii

The definition and the simplest properties of the Riemann zeta-function

1. 2. 3.

4.

5. 6.

7.

Definition of t (s) Generalizations of t (s) The functional equation of t (s) 1. The theta-series and its properties 2. Expression for the zeta-function in terms of the theta-series Functional Equations for L(s, X) and t(s, a) 1. Analytic continuation of L (s, X) to the region Re s > 0 2. Functional equation for e (r, X) 3. Functional equation for L(s, X) 4. Analytic continuation of t (s, a) to the region Re s > 0 5. Functional equation for t(s, a) Weierstrass product for t(s) and L(s, X) The simplest theorems concerning the zeros of t (s) 1. Consequences of the functional equation for t (s) 2. The theorem of de la Vallee-Poussin bounding the zeros of t(s) The simplest theorems concerning the zeros of L (s, X) 1. Consequences of the functional equation for L (s, X) 2. A de la Vallee-Poussin theorem for the zeros of L(s, X) 3. Page's theorems

8. Asymptotic formula for N (T) Remarks on Chapter I

Chapter II The Riemann zeta-function as a generating function in number theory

1. The Dirichlet series associated with the Riemann t -function 1. The function r (n) 2. The Euler function cp(n)

1 3 5 5 8

11 11 12 14 16 17 20 21 21 25 28 28 32 35 39 41

43 43 45

x Table of Contents

2. The connection between the Riemann zeta-function and the Mobius function 45 1. The Mobius inversion formula 45 2. Some other formulas 47

3. The connection between the Riemann zeta-function and the distribution cl~~oom~~ ~

4. Explicit formulas 51 1. Expression for 1/1 (x) in terms of the zeros of ~ (s) 51 2. Expression for 1/1 (x, X) in terms of the zeros of L (s, X) 53 3. Selberg's formula 55

5. Prime number theorems 56 6. The Riemann zeta-function and small sieve identities 60 Remarks on Chapter II 63

Chapter III Approximate functional equations

1. Replacing a trigonometric sum by a shorter sum 64 1. Statement of the fundamental theorem 65 2. Reducing trigonometric sums to ttigonometric integrals 65 3. Asymptotic value of a special type of trigonometric integral 71 4. Proof of Theorem 1 76

2. A simple approximate functional equation for ~(s, a) 78 3. Approximate functional equation for ~(s) 81 4. Approximate functional equation for the Hardy function Z(t) and its

derivatives 85 1. The zeros of Z(t) 85 2. A formula for e(t) 85 3. Formula for Z(k)(t) 87

5. Approximate functional equation for the Hardy-Selberg function F(t) 95 Remarks on Chapter III 100

Chapter IV Vinogradov's method in the theory of the Riemann zeta-function

1. Vinogradov's mean value theorem 101 1. Lemma on the distribution of prime numbers 101 2. Linnik's lemma 104 3. Recurrence formula for J(P; n, k) 106 4. Statement and proof of the mean value theorem 110

2. A bound for zeta sums, and some corollaries 112 1. Auxiliary lemmas 112

Table of Contents xi

2. Estimate for a zeta sum 113 3. A bound for s(s) for Res < 1 116 4. A bound for Is (s) I in a neighborhood of the line Re s = 1 117

3. Zero-free region for s(s) 119 4. The multidimensional Dirichlet divisor problem 120 Remarks on Chapter IV 123

Chapter V Density theorems

1. Preliminary estimates 126 2. A simple bound for N((}, T) 128 3. A modern estimate for N((}, T) 131

1. The first case of S (p ) 134 2. The second case of S (p ) 146

4. Density theorems and ptimes in short intervals 148 5. Zeros of ~(s) in a neighborhood of the critical line 150

1. Preliminary facts about summation of arithmetic functions 150 2. Estimate for a multiple trigonometric sum 154 3. Upper bound for the number of zeros of ~(s) near the ctiticalline 158

6. Connection between the disttibution of zeros of ~ (s) and bounds on 1~(s)l. The LindelOf conjecture and the density conjecture 161

Remarks on Chapter V 166

Chapter VI Zeros of the zeta-function on the critical line

1. Distance between consecutive zeros on the critical line 168 2. Distance between consecutive zeros of Z(k) (t), k :::: 1 176 3. Selberg's conjecture on zeros in short intervals of the critical line 179 4. Distribution of the zeros of ~ (s) on the critical line 200 5. Zeros of a function similar to ~(s) which does not satisfy the Riemann

Hypothesis 212 1. The case 1/2 < Res < 1 214 2. The case Res = 1/2 215

Remarks on Chapter VI 239

Chapter VII Distribution of nonzero values of the Riemann zeta-function

1. Universality theorem for the Riemann zeta-function 241 2. Differential independence of ~(s) 252

xii Table of Contents

3. Distribution of nonzero values of Dirichlet L-functions 1. Preliminary lemmas 2. Theorem on shifts of Dirichlet L-functions 3. Theorems on shifts of zeta-functions of number fields 4. Independence of Dirichlet L-functions 5. The zeros of l;(s, a)

4. Zeros of the zeta-functions of quadratic forms 1. Basic lemmas 2. Joint distribution of values of Hecke L-functions 3. Zeros of the zeta-functions of quadratic forms

Remarks on Chapter VII

Chapter VIII Qtheorems

1. Behavior of 1l;((J + it)l, (J > 1 2. Q-theorems for l; (s) in the critical strip 3. Multidimensional Q-theorems

1. Statement of the theorems Remarks on Chapter VIII

Appendix

1. Abel summation (partial summation) 2. Some facts from analytic function theory 3. Euler's gamma-function 4. General properties of Dirichlet series 5. Inversion formula 6. Theorem On conditionally convergent series in a Hilbert space 7. Some inequalities 8. The Kronecker and Dirichlet approximation theorems 9. Facts from elementary number theory 10. Some number theoretic inequalities 11. Bounds for trigonometric sums (following van der Corput) 12. Some algebra facts 13. Gabriel's inequality

Bibliography Index

255 255 268 268 269 271 272 273 279 283 284

286 290 305 305 324

326 327 338 344 347 352 358 359 364 372 375 380 381

385 395

r I I i .1 ! ! ,

I I , 1

I I !

1

I I I

\

Chapter I

The definition and the simplest properties of the Riemann zeta-function

In this chapter we define the Riemann l; -function and study its basic properties as a function of a complex variable.

1. Definition of 8) The Riemann zeta-function l;(s) can be defined by simple formulas in two ways: as a Dirichlet series, or as an Euler product. We shall use the first of these expressions as our definition, and then obtain the second as a theorem.

Definition 1. If s = (J + it, (J > 1, then the Riemann zeta-function l;(s) is defined by setting

00 1 l;(s) = ~-. L..,; nS

11=1

(1)

If (J :::: (Jo > 1, then the series on the right in (1) converges uniformly and absolutely, since

00 1 100 du 1 < - < 1 + - = 1 + --. - L nUo 1 u Uo (Jo - 1

11=1

Each of the terms in (1) is a holomorphic function of s, and so, by Weierstrass' theorem, the function l; (s) is holomorphic for Re s = (J > 1.

2 I. The definition and the simplest properties of the Riemann zeta-function

Theorem 1 (Euler product). For s = (J + it, (J > 1, the follovving identity holds:

( 1 )-1 ~(s) = II 1 - pS '

P

(2)

where the product on the right is taken over all prime numbers p.

Proof Let X :::: 2. We define the function ~x (s) by setting

( 1 )-1 ~x(s) = II 1 - ---;

P:"OX P (3)

Each of the factors on the right in (3) is the sum of an infinite geometric progression:

1 ) -I 1 1 00 1 - =1+-+-+ .. = -. pS pS p2s L pillS

111=0

By the condition in the theorem, each of these progressions is absolutely con-vergent, and so they can be multiplied term-by-term. Thus, the right side of (3) is equal to

(4)

where 2 = PI < ... < Pj, and PI, ... , Pj are all of the prime numbers up to X. We note that none of the terms on the right in (4) are equal to one another, since if we had

'm'. 1111 IIIj 111 1 P

j'}' PI ... Pj = PI

then it would follow from the fundamental theorem of arithmetic (unique fac-torization of integers) that ml = m~, ... , mj = mj. Moreover, again using the fundamental theorem of arithmetic, we see that every natural number n :'S X can be represented in the form

17 = p~'l ... p~'j, (5)

where the m I, ... , m j are nonnegative integers, and the representation in (5) is unique. Consequently, the right side of (4) takes the form

1 I 1 L nS + L nS' l/:"OX II>X

(6)

2. Generalizations of I';(s) 3

where the I in the second sum stands for summation over those natural numbers n > X whose prime divisors are all :'S X. We give an upper bound for this sum:

tl I 1 1 L nS :'S L nO- < L nO- :'S II>X II>X l/>X


product for ~ (s):

L(s, X) = II (1 _ X(~))-1 , P P

Res> 1. (2)

The proof of (2) is exactly the same as the proof of Theorem 1.1. If x(n) = Xo(n) is the principal character modulo m, i.e., if Xo(n) = 1 for (n, m) = 1 and Xo(n) = 0 for (n, m) > 1, then from (2) we obtain

( ( )) -I (1 )-1 L(s, Xo) = II 1 - XO ~ = II 1 - -----; p P (p.m)=1 P

= II (1 -~ ) II (1 -~ ) -I = ~(s) II (1 -~ ) . plm P p P pllII P

Thus, L(s, Xo) differs from ~(s) only by a simple factor. In many respects the two functions ~ (s) and L (s, X) have a lot in common.

The different functions L(s, X) (notice that for fixed m there are 1 we have

~(s) = A(s), L(s, X) = B(s, X), ~(s, a) = A(s, a). Lemma 1 below gives the functional equation of the theta-series, which is the basic ingredient in the construction of the functions A(s), B(s, X) and A(s, a).

1. The theta-series and its properties

Definition 1. For an arbitrary complex number a and a complex number r with positive real part, the theta-series e(r; a) is the function

00

e(r; a) = L exp (-ll'r(n + a)2) . 11=-00

This function e (r; a) is holomorphic in the half-plane Re r > O. In addition, it obviously satisfies the relation e(r; a) = e(r; a + 1).

Lemma 1. One has the following identity:

e (~; a) = -JTII~ exp (-ll'n2r + 2ll'il1a) = =-JTexp(-ll'a2/r)e (r;_i:). (1)

Proof We first prove the lemma when r and a are real numbers. It will then follow by the principle of analytic continuation of identities (Theorem A.9.2) that (1) holds for complex r and a as well. Thus, let r = x > O. Without loss of generality we may assume that 0 :::: a < 1. We take N > 10 and M = N 6 , and we consider the integral

10.5 sin ll'(2M + l)u 1(11)= . exp(-ll'x(n+a+u)2)du. -0.5 Slllll'U Using the relation

sin ll'(2M + l)u sin ll' u

M

L k=-M

e-2Tfiku ,


we easily find that

and

),0.5 sin n (2M + 1) u

------du = 1, -0.5 sin nu

I (n) = e-JLt (l1+a)2 + R(n),

R(I1) = ------ e-JfX (I1+a+lI) - e-JfX (Il+a) duo ),0.5 sinn(2M + l)u ( 2 2)

-0.5 sin nu

(2)

We find an upper bound for the absolute value of R (n) under the condition that - N :::: 11 :::: N. To do this we write R (n) as a sum of three terms

where N-3

II = 1- 1, and let r be the rectangular contour joining the vertices - X, X, -X + ik, X + ik. Then the Cauchy residue theorem gives

(3)

8 1. The definition and the simplest properties of the Riemann zeta-function

The last two integrals are each no greater in absolute value than

If we take the limit in (3) as X ---?- +00, we find the following value for our original integral:

Thus,

and N M L e-~(n+a)2 + 0 (N-I) = .JX L e-nxk2+2nika + 0 (e-~N) .

II=-N k=-M

Taking the limit here as N ---?- 00, we obtain the lemma.

Corollary 1. Set e (x) = e (x, 0); then for x > 0 e(x- I ) = .JX e(x).

2. Expression for the zeta-function in terms of the theta-series

Theorem 1. The following equality holds for Re s = (5 > 1:

1 100 n-s/2r (~) ~(s) = + (x s/2- 1 + x-s/2- 1/2) w(x)dx, 2 s(s - 1) 1

o

(4)

(5)

Proof We use the integral formula for the gamma-function (Theorem A.3.4). For Re s > 0 and for 11 a natural number we have

r (~) = 100 e-ll u s/2- ldu = nS 100 e-m/2'ns/2xs/2-ldx (we have made the change of variables u = nn2x). Consequently,

. n-s/2r (~) n-s = 100 e-mhxs/2-ldx.

3. The functional equation of ~ (s) 9

We now suppose that Re s = (5 > 1 and sum the last equality over all n. The result is

Changing the order of summation and integration, we obtain

= lim roo x s / 2- 1 (t e-nIl2X ) dx N~+ooJo 11=1 Next, for x > 0 we have

On the one hand, the last integral does not exceed

on the other hand, it is no greater than

Thus, we obtain the bounds


and so

Thus, we have the equality

n-s/2r (~) ~(s) = leo x s/2- 1w(x)dx. From (4), which in expanded form says that

eo eo L exp (_~n2) =,jX L exp (-nxn2) , 11=-00 /1=-00

we easily find the following functional equation for w (x): I 1,jX r:;

w(x- ) = -2 + 2 + V X w(x). Using this relation and then making the change of variables x ----+ X-I (in the second integral below), we obtain

leo x s / 2- 1W(x)dx = 11 x s/ 2- 1W(x)dx + leo x s/ 2- lw(x)dx = leo (X-s /2- IW(X- 1) +x s / 2- 1w(X)) dx

___ + (x s/ 2- 1 + X- s/ 2- 1/2) w(x)dx, 1 leo s(s - 1) I

as was to be proved. o

We note some corollaries of Theorem 1. Since w (x) = 0 (e- rrx ) as x ----+ +00, it follows that the improper integral on the right in (5) converges absolutely and uniformly in the half-plane Res> K for any K. Weierstrass' theorem (Theorem A.2.1) then implies that, as a function of the complex variable s, this integral is

4. Functional Equations for L(s, X) and C;(s, 0') 11

holomorphic in the entire s-plane. The relation (5) was proved under the assumption that Re s > 1. But the right side of (5) is defined for all s, i.e., this formula gives the analytic continuation of the function ~(s) onto the entire s-plane. We take the following to be the function A(s) mentioned at the beginning of the section:

A(s) = ns/2r- 1 (~) ( 1 + (eo (xs/2-1 + X- s / 2- 1/2) W(X)dX) . 2 s(s - 1) il

The gamma-function res) has a first order pole at the point s = 0; we also have r(/2) =,fii. Thus, A(s) is a regular function on the entire s-plane except for the point s = 1, where it has a simple pole with residue 1. Finally, it is easy to see that the right side of (5) does not change when s is replaced by 1 - s. We define a function ~ (s) by the formula

~(s) = lS(S - 1)n-s / 2r (~) ~(s). We have proved

Theorem 2. ~ (s) is an entire junction, and

~(s) = ~O - s).

The relation (7), or the equivalent relation

is called the functional equation of the Riemann zeta-function.

4. Functional Equations for L(8, X) and ((8, a) We now obtain analytic continuations of the functions L (s, X) and ~ (s, a).

1. Analytic continuation of L( s, X) to the region Re s > 0 The analytic continuation to Re s > 0 is achieved by the following lemma.

Lemma 1. Suppose that X (n) is a nonprillcipal character modulo m, and let

Sex) = L x(n). n:Sx

(6)

(7)

(8)


Then for Re s > lone has

L(s, X) = s lco S(x)x-S-Idx. (1)

Proof Let N :::: 1, Re s > 1. Using partial summation (Theorem A. 1. 1), we obtain

where c(x) = S(x) - 1. Since Ic(x)1 ::: x, if we take the limit as N ~ 00 we find that

as was to be shown. o

Corollary 1. Theformula (1) gives an analytic continuation of L(s, X) to the half-plane Re s > O.

Proof Since X =1= Xo, we have IS(x)1 ::: cp(m),

and so the integral on the right in (1) converges absolutely and uniformly for Res:::: 0"0 > 0 for any 0"0. Hence, by Weierstrass' theorem (Theorem A.2.l), this integral is a holomorphic function of s on the half-plane Re s > O. 0

2. Functional equation for O(r, X) To extend L(s, X) onto the entire s-plane, we first derive functional equations for two functions e(r, X) and el (r, X) which are variants on the function e(r).

Note that if XI is a primitive character modulo kl and X is the character modulo k, k =1= kl' that is induced by XI, then for Res> lone has the identity

(2)

Because of (2), we need only derive the functional equation for L(s, X) for X a primitive character.

4. Functional Equations for L (s, X) and ~ (s, a) 13

Lemma 2. Suppose that X is a primitive character modulo k. If X is an even character, define e(r, X) by setting

co

e(r, X) = L x(n) exp (-nrn2 / k) , r > 0; II=-CO

if X is an odd character, define el (r, X) by setting co

el (r, X) = L nx(n) exp (-nrn2 / k) , r > O. II=-CO

Then these functions e (r, X) and e1 (r, X) satisfy the following functional equations:

g(xWI (r, X) = i ,,)k.-3 e1 (r- I , :X), where g(X) is the Gauss sum

k

g(X) = L x(a) exp(2nia/ k). a=1

Proof We use Lemma 3.1 with s = r > 0 and with ex real: co co L exp (-n(n + ex)2r-l) = Jr L exp (-nn 2r + 2ninex) .

II=-CO II=-CO

We have the following chain of identities:

_ _ nrn 2:rrzmn k co ( 2 g(x)e(r, X) = ~ x(m) II~CO exp --k- + k )

k co ( = ~x(m),,)k,-III~ exp -lmr-1 (n + ;r)

_ (kn+m) k co ( 2 = ,,)kr-1 ~ x(m) I1~CO exp -n kr ) = ,,)k.- I mf;co x(m) exp ( - nk~2) = ,,)k.-1 e(r-1, X),

(3)

(4)

(5)

(6)

which proves (3). To prove (4), we differentiate (6) term by term and then replace r by rk- I and ex by mk- I (the series can be differentiated term by term because


the resulting series is uniformly convergent). As a result, in place of (6) we have

co (Jrrn 2 2Jrimn) . r,-;; ~ (kl1 + 11l)2) L 11 exp --k- + k = IV kc3 L-.t (kn+l11) exp -Jr kr . 11=~OO 1l=~OO

Proceeding as before, from this we obtain

k co (n 2 Jrr 2Jrimn) g(xWI(r, X) = Lx(m) L 11 exp --k- + k 111=1 11=-CO

~ k ~ ( (kl1 +111)2) = iv kr-3 Lx(m) L-.t (kn + 111) exp -Jr kr

111=1 11=-CO

co ( Jr112

) = i~kr-3 L nx(n) exp -kr

11=-00

. ~e ( -I -) = I V /(r- J 1 r ,X.

The lemma is proved.

3. Functional equation for L(8, X) Theorem 1. Let X be a primitive character modulo k, and let

0= {01', ifX(-1) = 1; if xC-I) =-1.

Define the function ~ (s, X) by setting

1 -(s+8)/2 (s + 0) ~(s, X) = (Jrk- ) r -2- L(s, X) Then one has the identity

o

Proof We repeat the argument used to prove Theorem 3.1. Suppose that X is an even character, i.e., X (-1) = 1. Then

4. Functional Equations for L(s, X) and ~(s, a) 15

We multiply both sides by X (n) and, assuming that Re s > 1, we sum over all natural numbers 11. We obtain

,,-'/2k'/2 r (~) L(." X) = f r,/2-1 (~X (n) cxp ( - Jr:n2) ) dr

Using the evenness of the character X, we have

~ (Jrr112) 1 L-.t x(n) exp --- = -e(r, X); k 2

11=1

(Jrk- 1f s/2 r (~) L(s, X) = ~ lco r s / 2- 1e(r, x)dr. We divide the last integral into two parts, make the change of variables r -----+

r- 1 in the first of the two integrals, and use (3). As a result we obtain

(Jrk- 1f s/2 r (~) L(s, X) = l lco r s / 2- l e(r, x)dr + ~ lco r-s / 2- 1e(r- 1, x)dr = ~ (CO r s / 2- l e(r, x)dr + ~~ (CO r-s / 2- 1/ 2e(r, X)dr.

2 il 2g(X) il (7)

The expression on the right is an analytic function for all s. Hence, the for-mula (7) gives an analytic continuation of L(s, X) onto the entire s-plane. Since r (s /2) -=f. 0, it follows that L (s, X) is everywhere regular. Next, if we use the fact that

g(X)g(x) = g(X)g(X) = k (since X( -I) = 1), we see that when we replace s by 1 - s and X by X the right side of (7) is multiplied by the factor .Jk / g (X). This gives us the theorem in the case 0 = 0.

Now suppose that X is an odd character, i.e., X( -1) = -1. We have

( l_I)-(S+I)/2r(S+1) -s lco (Jrrn2) s/2-1/2/ Jr j( -- n = n exp - -- r (, r. 2 0 k Consequently, for Re s > 1

(8)


The last expression gives an analytic continuation of L(s, X) onto the entire s-plane, and also shows that L (s, X) is everywhere regular. If s is replaced by 1 - s and X is replaced by X, the right side of (8) is multiplied by i~/g(X), where we use the fact that

g(X)g(x) = -k. This gives us the theorem in the case 8 = 1. The theorem is proved. o

We note that L (s, X) is an entire function. We next extend ~ (s, a) to the entire s-plane.

4. Analytic continuation of 8, ex) to the region Re 8 > 0 Lemma 3. The following equality holds for Re s > 0 and N E N:

N 1 1 ( 1 )I-S 100 p(u)du ~(s a) = '" + -- N + - + a + s , , L.. (n + a)S s - 1 2 N+l (u + a)s+1

n~ 2

where p(x) = x - [x] - 1/2.

Proof We take a natural number M > N and apply the Euler summation formula (Theorem A. 1.2). We obtain:

1 (M+~ du + S 1M+~ p(u)du (n + a)S - J N+l (u + a)S N+l (u + a)s+1 N+~


Using Lemma 3.1, for ro > 0 we have

/(s, a) = r s / 2- l e(r, a)dr + r s / 2- l e(r, a)dr l TO 100 o TO (0 1 100 = io r s / 2- 1_ L exp (-nI12r-1 + 2nina) dr + r s / 2- l e(r, a)dr o .jT IIEZ TO

= (0 r s/2- 3/2dr + (0 rs/2-3/2 L exp (-nn2 r- 1 + 2nina) dr+ io io II';iO

+ 100 r s / 2- l e(r, a)dr TO

2 (s-I)/2 100 ( ) = ro + r-s / 2- 1/2 ""' exp(2nina) . exp( -nn2r) dr+

s - 1 -1 L.,; TO II';i0

+ 100 r s / 2- l e(r, a)dr. TO (10)

The last expression is an analytic function on the entire s-plane. Moreover, from the representation (10) for / (s, a) it is clear that oVa / (s, a) is also an analytic function on the entire s-plane. If we let ro go to +00 in (10), we obtain for Res < 0:

where

/ (s, a) = L exp(2nina) 100 r-s/2- 1/2 exp (-nn 2r) dr litO

= L exp(2nina) roo (~) -s/2-1/2 e- T d (~) ~ nn nn litO (11)

= Lexp(2nina)nS/2-1/2Inls-lr (1 ;s) litO

= 2ns/2-1/2r (1 ; s) F(1 - s, a), 00

L cos2nna F(s,a) = . nS 11=1

The formula (11) gives an analytic continuation of F (s, a) and vaa F (s, a) onto the entire s-plane. For Re s > 2 we have

3 (s -1) ( 3 3 ) -/(s - 1, a) = n-(s-I)/2r -- -s(s - 1, a) + -s(s - 1, 1 - a) . 3a 2 3a 3a

4. Functional Equations for L (s, X) and l; (s, Q') 19

Since 3 00 3 00 -s(s - 1, a) = ""' -en + a)-s+1 = (1 - s) ""'(11 + a)-S = (1 s)s(s, a), 3a L.,; 3a L.,;

11=0 11=0

it follows that

33a

/(s - 1, a) = n-(s-I)/2r (y) (1- s)(s(s, a) -s(s, 1 - a) = -2n-(s-I)/2r (s: 1) (s(s, a) -s(s, 1 - a).

(12)

Because / (s, a) and aOa / (s, a) have analytic continuations onto the entire com-plex s-plane, we see from (9) and (12) that s(s, a) also extends analytically onto the entire s-plane. Solving (9) and (12) for s(s, a), we obtain

s(s,a) = l (n S/2r- 1 (~) /(s,a) -In(S-I)/2 r -1 C: 1) 33a

/(S -I,a).

For Res < 1, we obtain from (11):

s(s, a) = l (nS/2r- 1 (~) . 2ns/2- 1/2r (1; s) F(1 - s, a)-_ ~n(S-I)/2r-l (s + 1) . 2ns/2-1 r (2 - s) ~ F(2 sa)

2 2 2 3a '

=n ,_1/ 2r ((1-s)/2) fcos2nna +~nS-3/2 r(1-s/2) f2nsin2nl1a rcs/2) 11=1 n l-" 2 rs + 1)/2) 11=1 I1 I- s

(13) Using the doubling formula for the gamma-function (Corollary A.3.5), which

gives

r((1-s)/2) res /2)

and also

r((1 s)/2)r(1-s/2) 1. n c -2((1-s)/2) . = - sm -s2"" n2 r(1-s)

n/ sm(ns/2) n 2 '

r(1 - s/2) r(s + 1)/2)

r(1 s/2)r(1- (s + 1)/2) n / sin(n s /2 + n /2)

= ~ cos ~s . 2~ . r 2((1-S)/2) r(1 - s), we see from (13) that for Re s < 0 we have

r( ) 2(2 )S-lr(1 ) (. n Loo cos2nl1a n Loo sin2nl1a) ., s, a = n - s sm -s 1 ' + cos -s . 2 n -5 2 n l - s

11=1 11=1


This completes the proof of the theorem. o

5. Weierstrass product for ((8) and L(8, X) The Weierstrass preparation theorem (Theorem A.2.4), which is a generalization of the fundamental theorem of algebra, can be applied to the entire functions ~ (s) and

~(s, X). To do this we must first determine the order of growth of these functions.

Lemma 1. ~(s) and ~(s, X) are entire functions of order 1.

Proof By Theorem 3.2, we have ~(s) = ~(1 - s), and so it suffices to estimate 1~(s)1 on the half-plane Res ~ 1/2. According to the corollary to Lemma 4.3 with N = 1, for Res ~ 1/2 we have

Is(s)s(s - 1)1 = 0 (Isn . Furthermore, by Theorem A.3.5, one has

l1(s)1 .::; exp(clsllog lsI). Since also In-s / 2 1 .::; exp(ct!sl), it follows that

~(s) = }S(S - 1)n-s/ 2r (~) s(s) is a function of order at most 1. But as s ----+ +00 one has

log 1(s) ~ slogs, i.e., log~(s) '" slogs,

and so the order of ~ (s) is equal to 1. Repeating this argument word for word and using Theorem 4.1 and Lemma 4.1,

we find that ~ (s, X) is also an entire function of order 1. The lemma is proved. 0

If we use this lemma and the observation that as s ----+ +00

for c > 0 arbitrarily large but fixed, we can apply Theorem A.2.4 to conclude that each of the functions ~ (s) and ~ (s, X) can be represented in the form

00 sreA+Bs II (1 - s / PI1) eS / p" , (1)

11=1

where the series I::IIPIII- I diverges and the series I:~=llplIl-I-E converges for any E > O. Thus, the product (1) has infinitely many factors, and so Hs) and

6. The simplest theorems concerning the zeros of S (s) 21

~(s, X) have infinitely many zeros. In (1) the symbols A and B denote certain constants. One can show by a rather simple argument that the PI1 lie in the region o .::;Res .::; 1. Namely, let us take the case of ~(s). By (3.6) and (3.7), it is sufficient to show that ~ (s) =j:. 0 for Re s > 1, and this will follow if we prove that s (s) =j:. 0 for Re s > 1. From Theorem 1.1 we find that for Re s > 1

1 Is(s)1

1 II ( 1) ~ M(n) s(s) = 1 - pS = L.t -;;;-;

p 11=1

2:00 M(n) 2:00

1 /00 du (J -- < - < 1 + - - --'

n S n(J I u(J - (J 1 ' 11=1 11=1

(J-1 Is(s)1 > -- > O. (J

The same proof goes through for the function ~ (s, X).

From the formula (3.6) which defines ~(s) it is easy to see that ~(O) =j:. 0 and ~ (1) =j:. 0; and since 1(s /2) =j:. 0, it follows that all of the zeros of the function (1) are zeros of S (s). The same holds for ~ (s, X).

6. The simplest theorems concerning the zeros of ((8)

1. Consequences of the functional equation for ((s) The proofs of the simplest theorems on the zeros of S (s) are based upon Theorem 1.1, the formulas (3.6) and (3.7), and the representation of ~(s) in the form (5.1). That is,

Hs) = ~s(s - l)n-s / 2r (~) s(s) = eMBs fr (1 -~) eS/ p". 2 2 11=1 PI1

(1)

From Lemma 4.3 with Re s > 0 and N = 1 we have

(s) = ~ 1 s Joo p(u)du s 2 + 1 + s+1 S - I U We multiply both sides of this equality by s - 1 and take the limit as s ----+ 1. We obtain:

limes - l)s(s) = 1; s---+I

. 1 1 (1) 1 hm~(s)=--r - =-. HI 2 ~ 2 2


From Theorem 3.2 it follows that HO) = ~(l) = ) /2. Since (r(s) -I has the expansion

00

(r(s) r 1 = seY' II (1 + ;) e-s/ n , n=l

it follows that lim'---fo s r(s) = 1. Hence,

e A = ~(O) = -~(O) = 1/2, ~(O) = -1/2. Thus, we have found that ~ (0) = -1/2 and ~ (l) = 1/2.

Next, for Re s > 1 we easily obtain

00 1 00 1 ( 1-21-s)~(s) = ~ - -2~-L......t n' L......t (2n)'

n=l n=l 111

=1--+---+. 2' 3' 4"

The last series converges for all positive real s, i.e., for s > 0 we have

( l-S) 1 1 1 1 - 2 ~(s) = 1 - - + - - - + ... > O. 28 3' 4' This shows that ~(s) =f. 0 for s > O. Thus, the Pn in (1) are complex, i.e., the zeros of Hs), which are also zeros of ~(s), are not real numbers.

Besides the Pn, the function ~(s) has other zeros. In fact, by Theorem 3.2,

JT-s/2r (~) ~(s) = JT-(l-8)/2r (1; s) ~(1 - s). If Re s < 0, then Re(1 - s) > 1, and the right side does not vanish. But r (s /2) has poles at the points s /2 = -n, n = 1, 2, ... , i.e., ~ (s) vanishes for s = - 2n, n = 1,2, ....

Next, from (3.5)-(3.7) it follows that ~(s) = ~(s);

hence, whenever PII is a zero of ~ (s), so are 7511' 1 - Pn, and 1 - 7511" Thus, the zeros of ~ (s) - which are the complex zeros of ~ (s) - are symmetrically located with respect to the real axis 1m s = 0 and the line Re s = 1/2. We state these results in the form of a theorem.

Theorem 1. The zeros of the Riemann zeta-function ~ (s) are the even negative numbers -2, -4, ... , -2n, ... and the complex numbers PIl' which all lie in the strip 0 :s Re s :s 1 and are situated symmetrically with respect to the lines 1m s = 0 and Res = 1/2. The complex zeros of ~(s) are the zeros of the jimction ~(s).

6. The simplest theorems concerning the zeros of t; (s) 23

In the theory of the Riemann zeta-function the region 0 :sRe s :s 1 of the complex plane is known as the critical strip. The line Re s = 1/2 is called the critical line. The real zeros of ~(s), i.e., the numbers -2, -4, ... , -2n, ... , are called the trivial zeros of ~(s), and the complex zeros of ~(s) are called the nontrivial zeros.

Riemann conjectured that all of the complex zeros of ~ (s) lie on the critical line, i.e., Re Pn = 1/2 for all ll. As mentioned before, this Riemann Hypothesis is still unproved. In this section we shall explain the simplest qualitative and quantitative results concerning the Pn.

In what follows the complex zeros of the zeta-function will be listed in the order of increasing absolute value of the imaginary part (and when two or more zeros have imaginary part of the same absolute value, they are put in arbitrary order).

Theorem 2. The zeta-function satisfies the following relation:

~/(S) 1 00 (1 1) 00 (1 1 ) ------+ --+- + ---- +c ~ (s) - s - 1 ~ s - PII Pn ~ s + 211 2n ' (2)

where Pn runs through all of the complex zeros of~(s) and c is an absolute constant.

Proof The equality is proved by taking the logarithmic derivative of both sides of (1). 0

Theorem 3. Let Pn = f3n + i Yn, n = 1, 2, ... , be all of the complex zeros of ~ (s), and let T ::: 2. Then

00 1 L 2 < clogT. n=l 1 + (T - Yn)

Proof We take s = 2 + iT, divide the series in two parts, and estimate each part separately, as follows:

00 ( 1 L s+2n n=l 2~1 )

:s Cl log T.


We now multiply (2) by -1, take the real part of both sides of the equality, and estimate the result. We obtain

~'(S) (1 00 (1 1 )) f (1 1 ) - Re ~ (s) = Re s _ 1 - c - ~ s + 211 - 211 - Re 11= I ---;--p; + PI1

00 (1 1 ) :s c210g T - Re I: -- + - . 11=1 S - PII PII

Since for s = 2 + iT

\ ~'(S) I ~ A(n)

~(s) = ~ ll2+iT < C3,

it follows that

00 (1 1 ) ReI: --+-11=1 s - PII PII

:s c410g T.

We now note that 0 :s fJlI :s 1, and so 1

Re--=Re s - PI1 (2

1 2 - fJlI 0.5 fJlI) + i(T - YII) - (2 - fJlI)2 + (T - YII)2 ~ 1 + (T - YII)2'

R ~ - fJlI > 0 e - 2 2-' PII fJlI + YII This gives us the theorem. o

We list some corollaries which are often used in applications of the theory of the zeta-function.

Corollary 1. The number of zeros PII of the zeta-function for which T :s 11m Pili :s T + 1, is no greater than clog T. In particular, the multiplicity of a complex zero PII in the rectangle 0 :s Re s :s 1, 0 :s 1m s :s T, is bounded by 0 (log T).

Corollary 2. For T ~ 2 one has the bound 1 I: 2 = o (log T).

IT-y"I>1 IT - YI1I

Corollary 3. The following formula holds for -1 :s (J" :s 2, s = (J" + it: ~'(S) 1 1 - = -- + I: ---;=- + O(log(ltl +2)), ~(s) s - 1 II-y"ISI PI1

(3)

6. The simplest theorems concerning the zeros of S (s) 25

where the summation 011 the right is taken over all zeros PII of ~ (s) for which It - 1m PII I < 1.

Proof When s = (J" + it with 1:s (J" :s 2, we easily find that

(1 1) lsi :s I: - + - + I: 2 = O(log(ltl + 2)). n n n IIslll+2 11>111+2 Using (6.2), we obtain

~'(S) 1 00 (1 1 ) -=--+I: --+- + O(log(ltl +2)). ~(s) s - 1 11=1 S - PII PII

From this equality we subtract the same equality with s = 2 + it:

~'(S) 1 00 ( 1 1 ~(s) = -~ + ~ s - PII - 2 + it If IYII - tl > 1, then

PII ) + O(log(ltl + 2)).

1(J"+i~-PII - 2+i:-Pu!:S (:,~~)2:S (YI1~t)2' and (3) follows from Corollaries 1 and 2.

2. The theorem of de la Vallee-Poussin bounding the zeros of 8)

o

Theorem 4. There exists an absolute constant c > 0 such that the zeta-function has no zeros in the following region of the s-plane:

c Res=(J" > 1------ 10g(ltl + 2)

Proof Since ~ (s) has a pole at s = 1, there exists a positive number Yo such that ~(s) has no zeros in the region Is - 11 :s Yo. Let P = fJ + iy be a zero of ~(s); then clearly I Y I > Yo > O. Since for Re s = (J" > lone has

~'(S) ~ A(n) ~ --- = L-.t -- = L-.t A(n)n-a exp( -it log n), ~(s) 11=1 nS n=1

it follows that

~'(S) 00 -Re-- = I:A(n)n-a costlogn. ~(s) 11=1


For Yo, we obtain

C;'(o-+it) 00 (1 1) -Re . < B2log(ltl +2) - LRe -- + - ,

C;(o- + It) k=l s - Pk Pk where B2 > 0 is an absolute constant. The real parts 13k of the zeros Pk are nonnegative and at most 1 in absolute value; hence,

1 1 0- - 13k Re-- =Re = > 0;

S - Pk 0- - 13k + i(t - Yk) (0- - f3k)2 + (t - Yk)2

1 13k Re- = 2 2 ~ 0;

Pk 13k + Yk

-Re C;'(o- + it) < B 10 (It I 2) _ 0- - 13 C;(o-+it) 2 g + (o--f3)2+(t-y)2'

Replacing the last inequality by a weaker one and then substituting 2t in place of t, we find that

C;'(o- + 2it) -Re . < B2log(2ltl + 2). C;(o- + 21 t)

Now that the three terms on the right in (4) have each been bounded, we can bring the estimates together, obtaining

3 4(0- - 13) o ::s 0- _ 1 - (0- _ 13)2 + (t _ y)2 + B 10g(ltl + 2),

where B > 1 is an absolute constant. The last inequality holds for any t, I t I > Yo, and for any 0-, 1 < 0- ::s 2. We take

1 t = y, 0-=1+ ; 2B 10g(lyl + 2)

. 6. The simplest theorems concerning the zeros of ~ (s) 27

we then have 4 3

0- _ 13 ::s 0- _ 1 + B log(lyl + 2),

f3::s 1- (14Blog(lyl +2)r1, as was to be proved. o

From the theorem we easily obtain a bound for the logarithmic derivative of C;(s) slightly to the left of the line Re s = l.

Corollary 4. Suppose that T ~ 2 and c > 0 is the absolute constant in Theorem 4. Then the bound

holds in the region

C;'(o- + it) = 0(10 2 T) C;(o- + it) g

c 0->1------ 2log(T+2)' 2::s It I ::s T.

Proof Without loss of generality we may assume that 0- ::s 2. From (3) we have

1

C;'(o-+it)1 1 C;( .)::s L I 13 +cdogT.

o-+lt It-YIlI::;1 0-- n+i(t-Yn)1

Since f3n ::s 1 - clog-1 (T + 2) and 0- ~ 1 - c(2log(T + 2) r 1, it follows that

1

C;'(o-+it)1 2 C;(o- + it) ::s ~ 10g(T + 2) L 1 + c1log T = o (10g2 T),

It-YIlI::;1

as claimed. o

Corollary 5. One has:

Proof Let 0 < R = 3r < 1 and So = 1 + r + i to. Further suppose that r is such that for Res> 1 - R and to - 1 ::sIms ::s to + lone has


By Lemma A.2.2, we have the following inequality for Is. - sol :s r:

l' Ilog~(s) -log~(so)1 :s 2 (M -log I~(so)i) -- = M -log 1~(so)l, R -1'

where M = maxls-sol:'OR log 1~(s)l. Since we have

l SI I ~'(S) I M:s 10gl~(so)1 + - Idsl:S 10gl~(so)1 + O(Rlog2(ltl +2) So ~ (s) for some SI, lSI - sol = R, it follows that for Is - sol < r we have

log 1~(s)l-l :s Ilog 1~(s)l-log 1~(so)11 + Ilog 1~(so)11 :s Ilogl~(so)11 + O(Rlog2(ltl +2) :s log~(l + r) + O(Rlog2(ltl + 2).

(5)

If we use Corollary 4 and set r = cllog2(ltl + 2) with Cl a sufficiently small positive constant, then from the last inequality we find that

The corollary is proved. o

7. The simplest theorems concerning the zeros of L(s, X)

1. Consequences of the functional equation for L(s, X) The basic results concerning the zeros of L (s, X) are similar to those for the zeros of ~(s). Because of the identity (4.2), it suffices to consider L(s, X) with X a primitive character.

If X is an even character, then by Theorem 4.1 we have

If Res> 1, then L(s, X) has an Euler expansion:

L(s, X) = II (1 - X(r;)-1 p p

7. The simplest theorems concerning the zeros of L (s, X) 29

Hence,

1 = II (1 _ X(P) = ~ /-i(n)x(n) IL(s, x)1 pS ~ nS

p n=1

00 1 100 du 0"


Proof We again write out the expansion (5.1), recalling the above remark that the exponent l' is zero:

~(s, X) = eMBs IT (1 -~) eS / p". 11=1 PI1

(2)

Here A = A(X), B = B(X), and the Pn are all of the complex zeros of L(s, X) listed in the order of increasing absolute value of their imaginary part. We take the logarithmic derivative of both sides of (2) and use the functional equation for ~(s, X) from Theorem 4.1. We obtain:

and

~'(s, X) = B + f (~ __ 1_) , ~(s, X) 11=1 PI1 PI1 - S

~'(s, X) ~(s, X) =

~'(1 - s, X) ~(1-s,x)

Setting s = 0 gives

B(X) = ~'(O, X) = _ ~'(1,!) = -B(X) - f (~ +~). ~(O, X) ~(1, X) 11=1 PI1 1 - Pn

Since 1 - 75n is a zero of L (s, X) whenever PI1 is, and the numbers Pn and 1 - 75" have the same imaginary part, it follows that

00 ( 1 1 ) B(X) + B(X) = 2ReB(x) = - L =- + - ; 11=1 Pn p"

00 (1 1) 2 Re B(X) + L =- + - = 0; ,,=1 p" PI1

00 1 ReB(x) +Re L - = O.

,,=1 PI1

From (2) and the definition of ~ (s, X) we have

(JTk- 1) -(s+8)/2 r (s + 8) L(s, X) = eMBs IT (1 _ ~) eS / p". 2 ,,=1 p"

(3)

(4)

7. The simplest theorems concerning the zeros of L(s, X) 31

We now take the real part of the logarithmic derivative of the last equality. We find that -Re L' (s, X)/ L(s, X) is equal to 1 k -log--ReB(x) 2 JT

00 (1 1) Y 1 00 (1 1 ) -Re --+- ---Re---Re -- . L s - p" p" 2 s + 8 L s + 8 + 2n 2n

,,=1 n=1 For s = 2 + iT we easily obtain the bounds

00 (1 1 ) Re --L s +8 +2n 2n ,,=1

00 (1 1 ) ~ L s + 8 + 2n - 2n n=1

~ '" ~ + '" !:j L...... n L...... n2 ,,:'OT ,,>T

~cllogT;

I L'(s, X) \ 00 1 JT2

Re L(s, X) ~ ~ /1 2 = 6' If we also use (3), we find that

00 1 Re L ~ c210gkT;

2 + iT - PI1 ,,=1

1 1 0.25 Re = Re . 2: 2; 2+iT-Pn (2-{3n)+z(T-Yn) l+(T-y,J

00 1 L 1 + (T _ )2 ~ C3 log kT, 11=1 y"

as was to be proved. o

Keeping the notation and conditions of Theorem 1, we have the following corol-laries:

Corollary 1. The number of zeros p" for which T ~ 11m PI1 I < T + 1, does not exceed clog kT.

Corollary 2. One has the bound 1

'" ~ cl10gkT. L...... (T - v)2 IT-y"I>1 rl1


Corollary 3. The following relation 110lds for s = 0- + it, 1:s 0- :s 2, It I :::: 2: LI(s, X) """' 1 --- = L + O(logkltl). L(s, X) 11-1',,1:'01 s - PII

Proof It suffices to take the logarithmic derivative of both sides of (4), then from the resulting expression subtract the same relation with 2 + iT in place of s, and finally use Corollary 2. 0

2. A de la ValIee-Poussin theorem for the zeros of L(s, X) We shall prove an analogue for L(s, X) of de la Vallee-Poussin's theorem bounding the zeros of ~ (s). To do this we shall use the inequality 3 + 4 cos cp + cos 2cp :::: 0, where cp is a real number, and also the upper bounds for - Re LI (s, X) / L (s, X) when s = 0- + it, X = XI is a primitive character, and when s = 0- + 2i t, X = X? As mentioned before, the problem of bounding the real zeros of L (s, X) in the case when X is a real character presents special difficulties.

Theorem 2. Let X be a complex character modulo k, and let s = 0- + it. Then L (s, X) has no zeros in the region

c Re s = 0- > 1 - -----

- logk(ltl+2) (5)

If, on the other hand, X is a real character modulo k, s = 0- + it, then L(s, X) has no zeros in the region

c Res = 0- > 1 - ,

- logk(ltl + 2) It I > O. (6)

Proof We first treat the case when X is a primitive character. Suppose that X is a complex character, s = 0- + it, 0- > 1, t :::: O. Then we have

x(n) = exp(iw(n)),

I 00 00 L (s, X) """' A(n)x(n) """'

- = L = L A(n)n-U exp( -it logn + iw(n)); L(s, X) nS

11=1 11=1

LI(s, X) ~ -Re = LA(n)n-Ucos(tlogn-w(n));

L(s, X) 11=1

LI(o- + 2it, X2) ~ -Re . 2 = L A(n)n-U cos 2(t log n - w(n));

L(o- + 2lt, X) 11=1


3 (- LI(o-, xo)) + 4 (-Re LI(o- + it, X)) + (-Re LI(o- + 2it, X2)) :::: O. (7) L(o-, Xo) L(o- + it, X) L(o- + 2it, X2)

We find an upper bound for each term of this inequality, using the same argument as in the proof of Theorem 1. We have

LI(o-, Xo) ~ -u ((0-) 1 - = LXo(n)A(n)n :s --- < -- + CI;

L(o-, Xo) n=l ~(o-) 0- - 1

-Re LI(s, X) = -Re LI(o- + it, X) L(s, X) L(o- + it, X)

1 k 00 (1 1 ) = -log - - Re L -- + -

2 Tr S - PII PII 11=1

Y 2

1 00 (1 1 ) - Re-- - Re B(X) - Re L - -

s + 8 s + 8 + 2n 2n 11=1

00 1 :s c2logk(t + 2) - Re L --.

11=1 S - PII

If XI denotes the primitive character induced by X2, then Xl #- Xo, and

jL'(S,X2) LI(S,XI)j ,,"",p-Ulogp """' L(s 2) - L(s ):s L 1- -u :s L1ogP:S logk.

, X , XI plk P plk

(8)

Thus, if we apply (8) and recall that Re ( 1 / (s - PII)) :::: 0, n = 1, 2, ... , we find that

LI(o- + 2it, X2) LI(o- + 2it, XI) -Re L(o- + 2it, X2) :s -Re L(o- + 2it, Xl) + logk :s c3 1og k (t + 2).

We thereby obtain the inequality

3 1 -- - 4Re-- + clogkCt + 2) :::: O. 0--1 S-P (9)

Now suppose that P = f3 + iy is a zero of L(s, X). Without loss of generality we may assume that y :::: O. In (9) if we take t = y, 0- = 1 + (2clog k(t + 2) r1, we obtain

The first part of the theorem is proved.


We now prove the theorem for a primitive real character X. In the first place, we have

2 X = Xo, I L'(s, X2) ~'(s) I -----,--- - -- < log 1(. L(s, X2) ~(s)-

From Theorems 6.2 and 6.3 we find that ~'(s) 1

-Re-- < Re-- + c2log(t + 2). ~(s) s - 1

If we substitute these bounds into (7) and suppose that t = y, where p = f3 + i Y is a zero of L(s, X), we obtain

3 4 1 -- - -- + Re . + c4log key + 2) 2: 0; u - 1 u - f3 u - 1 + 21 y

4 3 u-l u - f3 ::: u - 1 + (u _ 1)2 + 4y2 + c4 log k(y + 2). (10)

We consider the case of "large" y and the case of "small" y separately. Suppose that y > K /log k, where K is an absolute constant, 0 < K < 1/ (5C4). Setting u = 1 + K/logk(y + 2) in (10), we find that

f3 ::: 1 - cs/logk(y + 2), Cs 2: 3/(5c4 + 16K-I ). Now suppose that 0 < y < K / log k. Using (8), we obtain

L'(u, X) ~ 1 2(u - f3) - < C2 log k - L.,; --- < C2 log k - ( 2 2 '

L(u, X) 11=1 u - PII U - f3) + y (11)

since the zeros P of L (s, X) have the form p = f3 i y. In addition, L'(u, X) ~ a ~ a

- L = L.,; x(n)A(n)n- 2: - L.,; A(n)n- = (u, X) 11=1 11=1

~'(u) 1 = -- > --- - C6 ~(u) U - 1

From this and (11) we have 2(u - f3) 1

---=-------=- < -- + C7 log k. (u - f3)2 + y2 U - 1 If we take u = 1 - A / log k and K = A /10 in the last inequality, we obtain

f3 ::: 1 - A/(1Ologk), This gives the theorem under the assumption that X is a primitive character modulo k. If X is an arbitrary character modulo k, then the theorem follows from the primitive case and the formula (4.2). D


3. Page's theorems

To prove the theorem bounding the real zeros of L (s, X) with X a real character, we need an auxiliary lemma.

Lemma 1. Let X be a primitive real character modulo k. Then c

L(1, X) 2: !1: 2' v k log k

Proof. For 1/2 ::: t < 1 we consider the function co

H(t) = Lalltll , all = LX(d). 11=1 dill

If n = pfl ... p~" is the prime factorization of 11, then II

all = IT (1 + X(P/,) + ... + X(p~')). /,=1

From this way of writing all it follows that all 2: 0 and alll 2 2: 1. We thus obtain the following lower bound for H (t):

co {CO (CO H(t) > Ltlll2 > i2 t ll2 du > in t Il2 du-2=

111=1 2

fi 1 -r======== - 2 > - 2 2J -log(1- (1 - t)) 2JI=t'

We now show that H(t) is nearly equal to L(l, X)/(l- t), i.e., we consider the difference

O(t) = H(t) _ L(l, X). 1 - t

We shall find an upper bound for the absolute value of Oct). First of all, we have

co( ) co co co ()II H Ct) = ~ ~ X (n) till = ~ X (n) ?; t rtl = ~ ~ ~ ;11 .

We next introduce the notation SII = L~=II X (m) / m and obtain co co co

_ L x(n)tll L(I, X) L til L x(n) til OCt) - -- - + (S - S -d -- - - . --I - til 1 - t 11 11 1 - t n 1 - t 11=1 11=1 11=1


00 (til til) 00 = ""' x(n) -- - - ""' SI+lt". ~ 1 til nO - t) ~

11=1 11=0

(2)

Each of the series SII is easy to bound using partial summation (Theorem A.l.l) and an estimate for a sum of character values (Theorem A.9.1O). For any M ~ m we obtain

ISIII ~ c-Vklogkln. Consequently,

00 00 t"+ 1 1 ""' SII+ I til < 2c-Vk log k ""' -- = 2c-Vk log k log --. ~ - ~n+l 1-t 11=0 11=0

(3)

Again applying an estimate for a sum of character values, we find that

00 (til t'l) Lx(n) 1-tll -n(l-t) = 11=1

00 ( Il ) ( til = ~ ~x(m) I-til t'l t

ll+1 tll+l) ---- +-----n(1 - t) 1- t ll+1 (n + 1)0 - t)

CI.JklOgkooj t'l t"+ 1 til O-t)tllj ~ I - t ~ 1 + t + ... + t"- I - 1 + t + ... + til - n(n + 1) - n + 1 cI.Jk log k 00 (til t ll+1 t'l)

< ""' - - + I - t ~ 1 + t + ... + t"-1 1 + t + ... + til n(n + 1) 11=1

1 < 2cI -Vk log k log --. 1 - t

From (2)-04) we obtain the following bound for IGU)I: 1

IG(OI < 3cI-Vklogklog--. 1 - t Thus,

L(1, X) = H(t) _ G(t) > 1 1 - t 2-/f=t

1 2 - 3cl-Vklogklog --. 1 - t

(4)

7. The simplest theorems concerning the zeros of L (s, X) 37

In this inequality we take t = 1- (coklog4 kr l with Co = (64(cI + 1)( We then obtain

I I _L_O_,_X_) > ~ v;;;k log2 k, 1- t 4

L(l, X) > . , 4 Fa .Jk log2 k

as was to be proved.

Theorem 3. Let X be a primitive real character modulo k. Then

Proof Let the real number a satisfy the inequalities 1

I---


L(s, XI) and L(s, X2) have real zeros /31 and /32, respectively. Then c

min(/3I, /32) < 1 - ---logkJk2

Proof The character x(n) = XI (n)X2(n) is a Dirichlet character modulo klk~. Since XI ::j: X2, it follows that X ::j: XO. For CJ > 1 we have

00

0:::; I:A(n)(1 + XI(n))(1 + X2(n))n~(J 11=1

~'(CJ) L'(CJ, XI) ----

~(CJ) L(CJ, xd From (11) we obtain

and also

< cllogkl

L'(CJ, X) L(CJ,X)

CJ - /31 ' 1

L'(CJ, xd L(CJ, XI) L'(CJ, X2) L(CJ, X2) < cllogk2 - --. CJ - /32

Substituting these bounds in (15), we find that

1 1 1 --+-- < --+C2 10g kl k2. CJ - /31 CJ /32 CJ - 1

If we now set CJ = 1 + (2c210g kJk2rl, then we conclude that

This completes the proof of Theorem 4.

(15)

o

Corollary 4. Let X run through all Dirichlet characters modulo k, and let L (s, X) be the corresponding L-functions. Then at most one of these L(s, X) can have a real zero /3 satisfying

c /3 > 1- --. logk

8. Asymptotic formula for N (T) 39

8. Asymptotic formula for N(T) We let N (T) denote the number of zeros of ~ (s) of the form s = P, 0 :::;Re p :::; 1, o :::;Im p :::; T. As we proved earlier, this is also the number of zeros of the function

in the rectangle 0 :::;Re s :::; 1, 0 :::;Im s :::; T. There is a rather precise asymptotic formula for N(T) as T ----+ 00.

Theorem 1. For T :::: 2 one has

T T T N(T) = -log - - - + o (log T),

2n 2n 2n (1)

where the constant implicit in the big- 0 notation is an absolute constant.

Proof First we suppose that there are no zeros of ~ (s) on the line 1m s = T. Let r be the rectangular contour in the s-plane with vertices s = 2 iT, s = -1 iT. Using Theorem A.2.11 and Cauchy's theorem, we see that

2N(T) = _1_ r ~'(s) ds. 2ni Jr ~(s)

By the definition of ~(s) and Stirling's formula (Theorem A.3.5), we have ~'(s) 1 1 1 1 r'(s/2) ~'(s) -- = - + -- - -logn + - . + --;

~(s) s s - 1 2 2 r(s/2) ~(s)

r'(s/2) s ( 1 ) r(s/2) = log "2 + 0 ~ .

We rewrite (2) as follows:

1 1T (r(2+it) ~'(-I+it)) 2N(T) = - - dt+ 2n ~T ~(2 + it) ~(-1 + it)

+ - - dCJ = h +h 1 12 (~'(CJ - iT) r(CJ + iT)) 2ni ~I ~(CJ - iT) ~(CJ + iT)

(2)

(3)

(4)

(5)

where we have let II and 12 denote the first and second integrals in the last formula. We first find an upper bound for Ihl. From (4) and (5) we have

1 12 (~'(CJ - iT) h=-2ni ~I ~(CJ - iT) ~'(CJ+iT)) ~(CJ + iT) dCJ + o (log T).


Next, by (6.3) we have


The problem of making this theorem effective - i.e., finding a lower bound for c (E) as a function of E, 0 < E < 1/2 - remains open.

7. One has the following theorem (see [45]-[46], [141]):

Let f(s) = G(s)/ pes), where G(s) is an entire function offinite order, pes) is a polynomial, f(s) = I:~1 a"n~S for Re s > 1, and I:~1 la"ln~(1+8) < 00 for any 8 > O. If

res /2)Jr~s/2 ~ b" f(s)['((1_s)/2)Jr~(I~S)/2 = ~ 1l1~s'

where "",00 Ib In~(1+8) < 00 fior any 8 > 0, then f(s) = c~(s), where c is an L.m=1 "

absolute constant.

Thus, in some sense the Riemann ~ -function is uniquely determined by its func-tional equation.

8. The asymptotic formula for N (T) was proved by Riemann [138].

Chapter II

The Riemann zetamfunction as a generating function in number theory

1. The Dirichlet series associated with the Riemann (-function Let {a,,} be a sequence of complex number indexed by the natural numbers. By the generating function of the sequence {a,,} we mean the formal Dirichlet series

00

L a" f(x) = -. nS ,,=!

(1)

In this book all of the sequences {a,,} considered will have the property that there exists a ero E R such that for er > ero

~ la,,1 < 00. L..t n ero, then all of the a" are zero. Namely, suppose that no is the least natural number for which a"o i=- O. Then

a"o = lim (j(er)ng) , 0'0. This implies that if two Dirichlet series give the same function in their domain of absolute convergence, then they have the same Dirichlet coefficients.

Let T (n) denote the number of natural numbers that divide n.

1. The function T(n) Let f (s) be the formal Dirichlet series defined by setting

f(s) = ~ T(n). L..t nS ,,=1

(3)

44 II. The Riemann zeta-function as a generating function in number theory

Note that the series s(s) = 2::1 n-s is absolutely convergent for Res> 1; hence, the following double series is also absolutely convergent for Re s > 1:

If we gather together the terms of the double series that have the same product mn, we find that

2 I:OO r(n) s (s) = -, /1" 1/=1

Res> l. (4)

We have thus proved that for any 8 > 0 the formal series (3) converges uniformly in the region Re s > 1 + 8, and the function given by this series coincides with S2(S).

From (4) we can obtain the following number theoretic corollary. By Theorem 1.1.1, for Re s > 1 we have

( 1 )-1

s(s) = II 1 - pS P

Hence,

( 1 ) -2 ( 2 3 ) S2(S) = II 1 - - = II 1 + - + + ... . pS pS p2s

p p

If we expand the parentheses and use unique factorization, we find that

A similar argument shows that

k I:oo Tk(n) S (s)= -, 17" 1/=1

(5 = Res> 1,

(5)

where Tk(n) denotes the number of natural number solutions of the equation Xl X2 .. Xk = n. Again using Theorem 1.1.1, we obtain the formula

al a2 am _ k + a, - 1 III ( ) Tk (PI P2 ... Pili ) - g k _ 1 '

where (~) denotes the number of combinations of a objects taken b at a time.

2. The connection between the Riemann zeta-function and the Mobius function

2. The Euler function 2 one has the relation

s(s - 1) = f cp(n) , S (s) nS 1/=1

(5 =Res > 2,

45

(6)

where cp(n) is Euler's function, i.e., the number of nonnegative integers less than n that are prime to n.

To see this, we write

r(s 1) 1 -s ( 2) ., - - - P - 1 - p-s 1 + .!!..- + L + . .. -

s(s) - II 1 - pl-s - II ( ) pS p2s -p p

~ I] (I + (I - ~) (;, + ;:, + .. ) ). This, together with the formula

give us (6).

2. The connection between the Riemann zeta-function and the Mobius function

1. The Mobius inversion formula

By Theorem 1.1.1, for Res> 1 we have the equality

s:S) = II ( 1 ;s ) . p

If we expand the parentheses in (1), we find that

1 I:oo fA-(n) s(s) = ---;;;-'

11=1 (5 = Res> 1,

(1)

where fA-(n) is defined as follows: fA-(1) = 1, fA-(n) = 0 if 11 is divisible by the square of a prime, and fA-(n) = (_1)k if n is the product of k distinct prime numbers. The function fA-(n) is called the Mobius function.


We now use the Riemann zeta-function to prove the so-called Mobius inversion formula.

Theorem 1. a) Suppose that f (n) and F (n) are two functions of natural numbers which are connected by the relation

F(Il) = L f(d). (2) dill

Then

f(n) = Lf-t (J) F(d). dill

(3)

b) If f (n) and F (n) are connected by the relation (3), then they also satisfy the relation (2).

Proof Let

Then for Re s > 1 it follows from (2) that

L F(n) L all S(S)N(S) = -- + -. I1S /1" I1:'ON II>N

(4)

If we multiply both sides of (4) by S~I (s) and then equate the Dirichlet coefficients for n :::: N, we obtain (3).

Similarly, if we start with (3), then we have S~I (s) ~ F(d) = ~ f(l1) + ~ bll

D ds D nS D 11" d:'ON II:'ON II>N

If we multiply both sides of this equality by S (s) and equate the corresponding Dirichlet coefficients, we obtain (2). 0

Remark. Theorem 1 could have been proved using formal infinite Dirichlet series. But in that case we would have had to impose restrictions on the rate of growth of f(n) and F(n), or else make an additional argument to justify our operations with formal Dirichlet series.

Theorem 2. The following formula holds:

L f-t(d) = {~: din

if II = 1, otherwise.

2. The connection between the Riemann zeta-function and the Mobius function 47

Proof Let f (n)= 1 for n = 1 and f (ll) = 0 for n =1= 1. Then in the notation of Theorem 1,

F(n) = L f(d) == 1. dill

Hence, by Theorem 1 we have

L f-t (J) = L f-t(d) = f(n), dill din

which is what Theorem 2 states. o

2. Some other formulas

We derive a few of the most commonly used formulas involving S~I (s) (0' > 1): s(s) = ~ !f-t(n)!, s(2s) D 11"

11=1

S3(S) = f r(n2) , s(2s) 11=1 nS

S4(S) = f (r(n) )2 s(2s) 11=1 nS

(5)

(6)

(7)

(8)

where v(n) denotes the number of distinct prime divisors of n. The proof of these formulas is based on the following general fact: if f (n) is a

multiplicative function, i.e., if

f (pal pa2 p~k) = f (p~l) f (p~2) ... f (p~k) ,

then


To prove (5), we note that

~ = II (1 - ~)-I II (1 -_1 ) = II (1 +~) = ~ 1/L(n)1 s(2s) p" p2s pS ~ nS .

p P P 11=1

The formula (6) follows from the relation

S2(S) ( 1 ) -2 ( 1) s (2s) = II 1 - p" II 1 - p2s =

p p

( 1 ) -I ( 1 ) - 1 - 1+- -- II ps II pS-P P

= II((1 +~) (1 + ~ + _1 + ... )) = pS V' p2s p

Next, we have on the one hand

~;~:~ = II (1 - ;s ) -3 II (1 - p~s ) = p P

= II (1 - ~)-2 II (1 +~) = P P p P

= 1]( (1 + ~,) (1 + :' + p~, + .. ) ) = = II (1 + ~ + ~ + ... + 2n + 1 + ... ) .

pS p2s pl1" p

On the other hand, since T(n) is a multiplicative function, we have

This proves (6).

3. The Riemann zeta-function and the distribution of prime numbers

Similarly,

Since formula (5) of 1 gives T2 (p~1 p~2 ... p~;,,) = (al + 1)2(a2 + 1)2 ... (alii + 1)2,

the equality (8) follows immediately.

3. The connection between the Riemann zeta-function and the distribution of prime numbers

49

Let rr (x) denote the number of prime numbers not exceeding x. By Theorem 1.1.1, for Res> 1 we have

( 1 )-1

s(s) = II 1 - -; , P P

and hence, using partial summation, we obtain

logs(s) = - '"""log (1 -~) = - roo log (1 -~) drr(x) = ~ p" il x" P

= -log (1 - ;s) . rr(x) I~ + /00 rr(x)dlog (1- xIs) = (1) /

00 rr(x)dx -s - I x(X S - 1)'

The identity (1) leads to one of the most important approaches to the study of rr(x). Namely, one can investigate the analytic properties of the function logs(s) and then find the corresponding inverse integral transform.


It turns out that the Chebyshev function 1/1 (x), defined below, is more convenient than Jr(x) from an analytic point of view. We first set

A(n) = {lOg p, 0,

Then we show that

if 11 = pk for some prime p; otherwise.

~'(s) _ f A(n) - ~(s) - -;;;-.

11=1

(2)

The function A (n) is called the von Mangoldt function, and the sum of the values of A(n) over all n ~ x is called the Chebyshev fUllction, i.e.,

1/I(x) = L A(n). To derive (2), we note that

~'(s) , -- = (-log~(s)) = ~(s)

n:::x

It is easy to see that all of the above operations with infinite series are justified, because of the uniform convergence of all of these series in the region Re s > 1 +8, where 8 is an arbitrary positive number.

We now obtain a formula similar to (1) which contains the function 1/1 (x). We have

Integrating by parts, we obtain

_ ~'(s) = x-S 1/1 (x) 100 _ roo 1/I(x)d(x-S). ~(s) 1 JI Since the definition of 1/1 (x) gives us 1/1 (1) = and 1/1 (x) = 0 (x log x) as x ----+ 00, we conclude that

~'(s) roo - ~(s) = S JI x-s - 11/I(x)dx. (3)

4. Explicit formulas 51

The formulas (1) and (3) point toward a deep connection between the distribution of prime numbers among the sequence of natural numbers and the properties of the Riemann zeta-function. If one wants to investigate the distribution of primes in an arithmetic progression, then the analogous role is played by Dirichlet L-functions.

Let q be a natural number, and let (a, q) 1. The generalized Chebyshev functions are the functions

1/1 (x; q, a) = L A(I1), II:"X, 11=([ (mod 'I)

and also the functions

1/1 (X , X) = L A(n)x(n), II:::X

where X is a Dirichlet character. By the definition of L(s, X) (see 1.1.2), we have

Hence,

_L'_(s_,X_) = (-lOgL(S,x))' = L(lOg (1- _X(_P)))' L(s, X) ps

p

~ X(p)logp 1 = D p" . 1 - X (p) / pS =

P

= ~ log p (X(P) + X(p2) + ... ) D pS p2s

p

= ~ A(n)x(n). D I1S 11=1

- ' = x-"d1/l(x,X) =s x- s - 11/I(x,X)dx. L'(s X) /00 /00 L(s,X) 1 1

This last formula is a natural analogue of (3).

4. Explicit formulas 1. Expression for 'Ij;(x) in terms of the zeros of (8)

(4)

(5)

In 3 we obtained a formula (see (3)) which expresses the logarithmic derivative of the Riemann zeta-function in terms of the Chebyshev 1/I-function. In this section we take up what in some sense is the inverse problem: finding an expression for the Chebyshev 1/I-function in terms of the zeros of the Riemann zeta-function.


Theorem 1. Let 2 :'S T :'S x. Then

1/r(x) = LA(n) = x - L xP

+ 0 (XlO;2 x) , n:'Ox IImpl:'OT p

where p runs through the zeros of the zeta-function in the critical strip. Proof By Theorem 1.6.2, for Re s > 1 we have

s'(s) 00 A(n) 1 00 (1 1) 00 (1 1 ) - s(s) = ~ ----;;;- = s - 1 - ~ s - PII + PIl - ~ S + 2n - 2n - Bo,

(1) where PIl lUns through all of the nontrivial zeros of s(s). By Theorem A.S.l, for b = 1 + log-j x we have

1 I b+iTI ( s'(S) x" (XlOg2X) 1/r(x) = -. -- -ds + 0 , 2m b-iTI s(s) s T

(2)

where T :'S Tj :'S T + 1 and Tj is taken in such a way that the distance from the line Ims = Tj to the nearest zero of s(s) is log-l T (this is always possible, since, by Corollary 1.6.1, the number of zeros of s(s) satisfying T :'SImp :'S T + 1 is o (log T). We consider the integral

J = _1 r (_ s'(S) X S ds, 2:rri Jr s(s) s

where r is the rectangle with vertices at the points biTj, -O.SiTl (see Fig. 1). Using Cauchy's Theorem, from (1) we obtain

J - x _ L x P _ S' (0) - IImpl:'OT1 P s(O) .

(3)

1 + Tl

r n

_1 0 1 b 2

ill -Tl

Figure 1

3. The Riemann zeta-function and the distribution of prime numbers 53

We now estimate the integrals over sides I, II and III of r. The integrals over sides I and III are equal in absolute value, and they are bounded by

- --- -ds < - dCJ. 1 IJO.5+iTI ( S'(S) XS I X Jb \ s'(CJ + iTj) I 2:rr -O.5+iTI s(s) s T, -0.5 s(CJ + iTl) (4)

The integral over side II does not exceed

1 IJ-O.5+iTI ( s'(S) X S I 1 JTlls'(-O.S+it)1 dt - - -- -ds < - . . (5) 2:rr -0.5-iTI s(s) s -.Ji -TI s(-O.S+lt) log-jx+ltl

We now estimate the expression Is'(CJ + it)/s(CJ + it)l, where either -0.5 :'S CJ :'S band t = Tl, or else CJ = -0.5 and 2 < It I :'S Tj. Once again, we have (by Corollary 1.6.3)

s'(CJ + it) s(CJ + it) L + O(log(ltl + 2). It_Ynl:SlCJ-CJIl+i(t YIl)

The last sum has order o (log2 x), since if It I :'S T" then CJ = -0.5, and the number of zeros of s(s) for which It - YIlI :'S 1 is no more than o (log(ltl + 2); on the other hand, if t = T j , -0.5 :'S CJ < b, then by our choice of T j we have

This estimate, along with (4), (5) and (3), implies the theorem. o

2. Expression for 'l/J(x, X) in terms of the zeros of L(s, X) For the generalized Chebyshev 1/r-function one has a similar formula expressing this function in terms of the zeros of the cOlTesponding Dirichlet L-function.


Theorem 2. Let X be a primitive character modulo k, 2 ::s k ::s T ::s x, Theil

xP

- kP (x log2 x) 1f; (x, X) - 1f; (k, X) = - L + 0 T ' IlmplST P

'where P runs through the nontrivial zeros of L(s, X),

Proof By Corollary 1.7.1, there exists T1, T ::s T1 ::s T + 1, such that I Tl - 1m Pili > ~ log kT, where PII runs through the zeros of L (s, X), We consider the rectangle r with vertices at the points b i T1, -0,5 i T1, where b = 1 + log -1 x, Integrating

x d -, -(logL(s, X)) s ds

over the contour r, we find that

-1-1 (_ LI(S,X)) XS -kS ds = _ x P -kP 2 ' ~ +e logx, m r L(s, X) s L P

IImplSTJ

where lei ::s 1. By Theorem A.5,1 (with ex = 1, x = N + 0,5),

1f;(x, X) = -, - -ds + 0 , 1 lb+iTI ( LI(S,X)) XS (XIOg2X) 2m b-iTI L(s, X) s T

We now estimate the integrals over the upper, lower and left sides of r, The integrals over the upper and lower sides of r have the same bound, Using Corollary 1. 7,3 and our choice of T1, we obtain

1 Ib+iTI ( LI(S, X)) x S - - -ds < 2Jri , -O.HiTI L(s, X) s -

< - -dO' e jb ILI (0'+iT1,X)1 x

- 2Jr -0.5 L(O' + iTl, X) T1

e jb 1 = - L ' + O(logkT)

2Jr -0.5 ITI-y"ISl 0' - 0'11 + I(T1 - YII)

~ 0 (x IO~ kT) ~ 0 (x 10;' x ) . x -dO' T1

~ '''I'

I 3, The Riemann zeta-function and the distribution of prime numbers

The integral over the left side of r is bounded as follows:

I_I JTI (_ LI(-O,5 + it, X)) X-O.Hit dtl 2Jr -TJ L(-O,5+it,X) -O,5+it

_osjTIILI(-O,5+it,X)1 dt ::sx . -TI L(-O,5+it,X) O,5+ltl

_ (IOg2 kT) -0 r:: ' yX

since, by Corollary 1.7,3,

I LI(-O,5 + ~t, X) I = 0 (logk(ltl + 2)), L(-O,5 + It, X)

Combining the above bounds, we obtain the theorem,

3. Selberg's formula

55

o

We now prove another identity which relates the zeros of the Riemann zeta-function to the Mangoldt function,

Theorem 3. Set AxCn) = A(n) if 1 ::s n ::s x, and set Ax(n) = (log-1 x)A(n) log(x 2 / n) if x ::s Il ::s x 2, Then

S/(S) AxCn) x 2(l-s) - x 1- s s(s) = - L ---;;;- + (1 - s)2logx +

I1


By Lemma A.5.2, we then obtain

A(n) (x X2) A(n) (X2) AAn) I = - ~ -.- log - - log - - ~ - -log - = log x ~ --. ~ liS 11 11 ~ nS 11 ~ /1" n.:Sx x


we obtain equation (2) of Theorem 1 from (4) and the above relations. This com-pletes the proof of the theorem. 0

We now take up the question of the distribution of primes in an arithmetic pro-gression.

Theorem 2. Let (k, I) = 1, x > 1. Then x xfh XI (I)

1jr(x; k, I) = cp(k) - EI f3ICP(k) + O(x exp(-coVlog x)), (6)

and

(7)

where E I = 1 if there exists a real character XI modulo k for which L (s, xd has a real zero

and E I = 0 otherwise.

1 f31 > 1---log2k'

Proof If k > exp(,Jlogx), then (6) and (7) obviously hold. Suppose that k ::s exp( ,Jlog x). By Corollary I.7.4, we have

1jr(x) EI 1 1jr(x; k, I) = cp(k) - cp(k) XI (l)1jr(x, XI) + cp(k) L 1jr(x, xYx(l) + o (log2 x).

X;iXQ,XI

If X =j:. xo, XI and X* is the primitive character modulo kdk which corresponds to X, then, setting T = exp(,Jlogx), by Theorem 4.2 we have

'" xp

1 1jr(x, X*) = 0 A(n)x*(n) = - L - + 0 (x exp( --Vlog x )). I1:"X ilmpi:"T, Rep>O p 2

By Theorem I.7.2, CI C

Rep < 1- -- < 1 - logkT - ,Jlogx'

Hence, by Corollary I. 7.1,

11jr(x, x*)1 ::s x exp(-coVlog x). Consequently,

1jr(x, X) = 1jr(x, X*) + o (log2 x) xexp(-coVlog x).

-------------- ---- '"

5. Prime number theorems

By a similar argument, for X = XI we find that

Thus,

1jr (x) 1jr(x; k, I) = cp(k)

XI (I) Xf31 E I -- + O(xexp(-coVlog x)). f31 cp(k)

If we now use (1), we obtain the relation (6).

Next, we have

lX+o 1 ;rex; k, I) = --d1jr(u; k, I) + cJX log2 x). 2-0 log u Integrating by parts, we obtain

l X+O 1 l X 1jr(u; k, I) 1jr(x + 0; k, I) --d1jr(u; k, I) = du + -'------~ 2-0 log u 2 U log2 U log x 1 t' du XI (I) r u f31 - l du

= cp(k) J2 log2 u - EI f3ICP(k) J2 log2 u + X XI (l)x f31

+ (k)l - E lf3 (1)1 + O(xexp(-c~Vlogx)) cp og x I cp i( og x

59

1 X (I) l x U f31 - 1 = (k)Lix - EI~(k --du + O(xexp(-c~Vlogx)).

cp cp ) 2 log u

This proves (7), and with it Theorem 2. o


6. The Riemann zeta-function and small sieve identities We begin this section by proving a useful identity.

Theorem 1. Let 1 :::: U < N, where N is a natural number. Then for any complex-valued function f (x) one has the identity

2: A(n)f(n) = 2:/-L(d) 2: (logl)f(ld)-U 1 we have

0= t'(S) - t'(S) t(s) = (2: /-L(n)) (((S) - t'(S) . t(S)) = t(s) IlSoU nS t(s)

(2: /-L(n)) (s) - t'(S) . t(s) (2: /-L(n)) . U nS t(s) U nS 11~ Il~

(2)

Let

t(s) (" /-L(n)) = ~ bll L...t nS L...t nS 11 SoU 11=1

We note that bl = 1, bll = I:dll1, dSoU /-L(d), and hence, by Theorem 2.2, bl1 = 0 for 1 < k :::: U.

6. The Riemann zeta-function and small sieve identities 61

From (2) we now conclude that

0= (2: /-L::1)) (- f l~~n) + (2: A,;;1)) (f ':$) (2: /-L,;:l)) IlSoU 11=1 I1SoU 11=1 IlSoU

( " /-LCd)) (~lOgl) + (" ACn)) (~~) (" /-LCd)) + L...t ds L...t Is L...t nS L...t rS L...t dS dSoU 1=1 11 SoU 1'=1 dSoU

+ (" AC.n)) (1 + " bill) . L...t 11 S L...t m S I1>U III>U

From this it follows that

2: bill 2: A(I1)(nm)-s. U


Since

( '" !J,(ll)) 1 '" bll ~(s) L..t - = + L..t-,

I1 S 11 11:'0 V n>N

where bn = 2.:.d:'OV, 1111 fJ,(d), it follows that

Hence,

L bn L fJ,(m)(nm)-s. V

Chapter III

Approximate functional equations

In many applications of the Riemann zeta-function it is important to know the order of growth of !t(CT +it)! as t ----+ +00. If CT > 1, then !t(CT +it)! is bounded by a constant. If CT < 0, then from the functional equation (8) of I.3 and Stirling's formula it follows that

!t(CT + it)! = 0 (t l / 2-0-) . Thus, it remains to treat the case when 0 ::s CT < 1, i.e., we want to study the

behavior of !t(CT + it)! as CT + it goes to +00 along a vertical line in the critical strip. We first prove several auxiliary results, which themselves are of independent interest and have numerous applications both in analytic number theory and in related fields of mathematics.

1. Replacing a trigonometric sum by a shorter sum We shall call finite sums of the form

S = L cp(x) exp(2ni f(x) . a

66 III. Approximate functional equations

Then for any!::'" E (0, 1) one has the equality

~ ~(x)exp(2nif(x)) =

~ lb ~(x) exp(2ni (J(x) - /Ix) )dx + 0 (H 10g(f3 - ex + 2)), a - 6. ""Wej) + 6. a

(5) where ex = f'(a), 13 = f'(b), and the constant in the big-O depends only on!::"'.

Proof We suppose that b - a > 10, since otherwise the lemma is trivial. First suppose that ~(x) = l. We take the natural number 111 = [lOU2 K2], K = 1 + lexl + 1131, and for [a] + 2::: M ::: [b] - 1 we consider the integral

Since

10.5 sin (2m + 1)nx WM = . exp(2nif(M +x))dx. -0.5 SIll nx sin (2m + 1)nx

sinnx

III

~ exp(2ninx), 11=-111

it follows that

10.5 sin (2m + l)n X . dx = 1, -0.5 sIllnx and hence

where

10.5 sin(2m + 1)nx ( ) VM = . exp(2nif(M + x)) - exp(2nif(M)) dx. -0.5 SIll nx

(6)

We bound I V M I from above. To do this we represent V M as a sum of three integrals:

1

1/111 1-1/111 jl/2 VM = + +

-1/111 -1/2 1/111

We bound the first integral by applying the tangent line approximation to the expression in the large parentheses:

11/1ll 11/111 If'llxldx K 1 II -U -1/111 -1/111 X m

1. Replacing a trigonometric sum by a shorter sum 67

The second and third integrals are similar to one another, so we shall only show how to bound the third. We first integrate by parts:

j l/2sin(2m + 1)nx ( ) . . exp(2nij(M + x)) - exp(2nif(M)) dx = 1/111 SIlln~\

where

exp(2nif(M + x)) - exp(2nif(M)) sinnx

cos(2m + l)nx \1/2 . + (2m + l)n 1/111

j l/2 cos(2m + 1)nx + Y,dx, 1/111 (2111 + l)n . exp(2nif(M + x)) . 2nif'(M + x)

sinnx (exp(2nif(M +x)) -exp(2nif(M)))ncosnx

sin2 nx

The first term has order of magnitude 0 (K m -I) = 0 (U -I) . Next, we have

j l/2 cos (2m + l)nx K I 1 ------Y,dx - ogm -. 1/111 (2m + l)n' III U Thus,

WM = exp(2nif(M)) + 0 (U- l ). If we sum the last relation over M and use the definition of W M and (6), we obtain

[b]-1 ~ exp(2nif(x)) = L WM + 0(1)

[b]-I 0.5 111 = L 1 ~ exp(2ni (J(M + x) - nx) )dx + 0(1)

M=[al+2 -0.5 11 =_111

III [bj-l M+O.5 L L 1 exp(2ni (J(x) - nx) )dx + 0(1) 11=-/1/ M=[a]+2 M-O.5

111 L III + 0(1), l1=-Jll

(7) where

j [b1-O.5 III = exp(2ni (J(x) - /Ix) )dx. [a]+1.5


We now find a bound for the sum of the terms in (7) with n < f'ea) - b.. and n > !' (b) + b.., i.e., for the sum I; given by

I;= -1Il:S1I(_1)11 L !'(bl) _ n = 0(1),

( 1)11 "'" - = 0(1). L.. !'(al) - n

Next, for -m ::s 11 < !' (a) - b.. the ratio

increases monotonically. Hence,

We similarly bound the sum of these ratios for f'(b) + b.. < n < m. As a result we have

I; = 0 (log(,8 - a + 2) ) ;


L exp(2lTif(x) = L III + o (log(,8 - a + 2). a


for y + 6. < 11 :::; f3 + 6., it follows that

j ll C(u) = 'L exp(2JTi (I (x) - nx) )dx + 0 (log(f3 - ex + 2)), fX~/::":SIl:sf3+/::" a

and

'L


it follows that the first term on the right in the above equality has order of magnitude o (H AU-I), and the numerator of the integrand in the second term on the right is bounded in absolute value by the expression

Thus, the entire integral is 0 (H AU-I). We now compare 1 and 11, where

rb- c flex + c) exp(2Jl'if(x + c) 11 = io V2f"(c)(j(x + c) _ f(c) dx.

We take the difference 1 - h and integrate by parts. We obtain:

1 - 11 =

1 ( 1 1). = -. , - exp(2Jl'lf(x + c)

2m f (x + c) V2f"(c) (j(x + c) - f(c) b-c

+ o

(l b - C !"(x+c) f'(x+c) ) + 0 - 2 + 32 dx . o (j'(x+c) J8f"(c)(j(x+c)-f(c) / Using the formulas

we easily find that the first term in the expression for 1 - 11 has order of mag-nitude 0 (AU-I), and the integrand in the second term has order of magnitude o (AU-2 ). Consequently,

1 = 11 + 0 (AU-I) .

I. Replacing a trigonometric sum by a shorter sum 73

We now compute 11. We first make the change of variable f (x + c) - f (c) = u in the integral. Letting A denote the difference feb) - f(c), we have

exp(2Jl' if(c) lA exp(2Jl'iu) 11 = du

J2f"(c) 0 0t exp(2Jl'if(c) loo exp(2Jl'iu)

--=-----du J2f"(c) . 0 0t

exp(2Jl'if(c) 100 exp(2Jl'iu) --=--;:o::---d u .

J2f"(c) A 0t We find two different bounds for the last integral. First, we have

---:=---du < - + du 1. 1100

exp(2Jl'iu) I lHI du 1100 exp(2Jl'iu) I A 0t - A 0t HlJU

Second, for A > 0 we have

Since I

roo exp(2Jl'iu) dU\ _1_ = 1 . iAJU .J):. J feb) - f(c)

feb) - f(c) = ~ !,,(~)(b - C)2 (b - c)2 A -I, 2

If'(b)1 = If'(b c + c)1 = !"(~I)(b - c) (b - c)A- I, it follows that

1 VA 1 -r=;;~=~ -- . J feb) - f(c) b - c If'(b)IJA

We thus obtain

1 = exp(2Jl'if(c) roo exp(2Jl'iu) du + 0 (min (If'(b)I-1 -lA)). I J2f"(c) io 0t "

I b ep(c) exp(2Jl'if(c) 100 exp(2Jl'iu) = du+ C J2f"(c) 0 0t + 0 (H AU-I) + 0 (H min (If'(b)I-1 ,JA)) .

We similarly compute the first integral in (8), obtaining

1c ep(c) exp(2Jl'if(c) 100 exp(2Jl'iu) = du+ a .J2f" (c) 0 0t + 0 (H AU-I) + 0 (H min (If'(a)I-I, -lA)) .

Since

100 cos 2Jl'U -100 sin 2Jl'u _ 1 0t du - 0t du - -, o U 0 u 2


the lemma follows from the above relations. o

Corollary 2. If a trivial bound is used for the right side of the formula in the lemma, then one obtains

lib ep(x)exP(2nif(x))dxl HfA.

Remark. This bound holds with weaker restrictions on the functions ep(x) and f(x). Namely, it suffices to require that !"(x) and epl(X) be continuous on [a, b] and that

ep(x) H, a.:': x.:': b.

Proof of CoroUmy 2. Without loss of generality we may assume that b - a :::: 4-VA. If the root e of the equation fl (x) = 0 lies in the interval [a, b], then we write

l b 1c-JA 1 c+JA 1b a = a + c-JA + c+JA and find bounds for each of these three integrals. We use a trivial bound on the middle integral:

1

c+.JA HfA.

c-JA

The first and third integrals are similar to one another, so we shall only go through the derivation of a bound for the first. We may suppose that a < e--VA. Integrating by parts, we find that

1

c-.JA 1 ep(x) IC-.JA

ep(x) exp(2nif(x) )dx = - .. -- exp(2ni f(x)) -(/ 2m f'(x) a 1 l c-.JA epl(x)fl(X) - ep(x)!"(x) .

- -. 2 exp(2n1 f(x))dx. 2m a (f1(X))

Since fl (x) is a monotonically increasing function and fl (e) = 0, a .:': e .:': b, it follows that

1 11'(a)l:::: Ifl(e -JA)I = Ifl(e) - fA!"(~)I -VA'

so that the first term in the last formula has order of magnitude 0 (H -VA). The second term in the last formula is an integral which we bound trivially, using the fact that

1'(x) = fl(e + x - e) = !"(O(x - e).


We find that

1

c-.JA 1c-.JA HV-IA-IV + HA- l

-2. 2 dx HfA. a a A (x - e)

On the other hand, if e does not lie in the interval [a, b], then we write our integral in the form

1

b 1(/+.JA 1 b-.JA 1b = + + .

a a a+.JA b-.JA

The first and third integrals on the right have the trivial bound H -VA. To find a bound for the second integral, we integrate by parts:

1

b-.JA ep(x) Ib-.JA

ep(x) exp(2nif(x) )dx = . I exp(2ni f(x)) a+.JA 2mf (x) (/+JA

1 1 b-.JA epl(x)fl(X) - ep(x)!"(x) . - -. 2 exp(2m f(x) )dx.

2m a+.JA (fl (x))

If fl (x) > 0 for a .:': x .:': b, then

1 i"(b - fA) > fl(a +JA) = 1'(a) + fl/(~)fA lA' . yA

fl(X) = 1'(a +x - a)>> A-l(x - a).

Thus, the first term above has order of magnitude o (H./A). The second term is an integral which we bound trivially:

1

b-.JA 1b-.JA (HU-IA HA)

. + 2 dx a+.JA (/+.JA x - a (x - a)

H V-I A log(b - a) + H fA HJA.

If fl(X) < 0 for a .:': x .:': b, then we have the same type of bound. The corollary is proved. 0


4. Proof of Theorem 1

First suppose that Ilf'(a)1I =f. 0, IIf'(b)1I =f. O. In Lemma 1 we take /:),. = 0.5. Then L

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