Karatsuba Basic Analytic Number Theory Anatolij A

Karatsuba Basic Analytic Number Theory Anatolij A. Karatsuba

Translated from the Russian by Melvyn B. Nathanson

Springer-Verlag Berlin Heidelberg GmbH Anatolij A. Karatsuba Steklov Mathematical Institute, uI. Vavilova 42 117966, Moscow, Russia

Melvyn B. Nathanson School of Mathematics, Institute for Advanced Study Princeton, NJ 08540, and Department of Mathematics, Lehman College (CUNY) Bronx, NY 10468, USA

Title of the Russian edition: Osnovy analiticheskoj teorii chisel, 2nd edition, Publisher Nauka, Moscow 1983

Mathematics Subject Classification (1991): 11-01, 11 M06, 11 N05, l1N13, l1P05, 11P21, l1P32, l1P55

ISBN 978-3-642-63436-9

Library of Congress Cataloging-in-Publication Data Karatsuba, Anatolii Alekseevich. [Osnovy analiticheskoi teorii chisel. English] Basic analytic number theory/Anatolij A. Karatsuba; translated from the Russian by Melvyn B. Nathanson. p. cm. Translation of: Osnovy analiticheskoi teorii chisel. Includes bibliographical references and index. ISBN 978-3-642-63436-9 ISBN 978-3-642-58018-5 (eBook) DOI 10.1007/978-3-642-58018-5 1. Number theory. 1. Title. QA241.K3313 1993 512'.73---<1c20 91-40715

This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law ofSeptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993 Softcover reprinl of the hardcover 1sI edition 1993 Typesetting: Macmillan India Limited, Bangalore 560 025 41/3140-543210-Printed on acid-free paper Contents

Translator's Preface ...... viii Introduction to the English Edition ...... ix Introduction to the Second Russian Edition...... x Notation...... Xll

Chapter I. Integer Points...... 1 § 1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results ...... 1 §2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums...... 6 §3. Theorems on Trigonometric Sums...... 10 §4. Integer Points in a Circle and Under a Hyperbola...... 21 Exercises ...... 25 Chapter II. Entire Functions of Finite Order...... 27 §1. Infinite Products. Weierstrass's Formula...... 27 §2. Entire Functions of Finite Order ...... 32 Exercises...... 38 Chapter III. The Euler Gamma Function ...... 41 §1. Definition and Simplest Properties...... 41 §2. Stirling's Formula...... 44 §3. The Euler Beta Function and Dirichlet's Integral...... 45 Exercises ...... 48 Chapter IV. The Riemann Zeta Function...... 51 §1. Definition and Simplest Properties...... 51 §2. Simplest Theorems on the Zeros...... 56 §3. Approximation by a Finite Sum...... 61 Exercises ...... 62 Chapter V. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series...... 64 § 1. A General Theorem ...... 64 §2. The Prime Number Theorem...... 66 vi Contents

§3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function...... 69 Exercises ...... 70

Chapter VI. The Method of I.M. Vinogradov in the Theory of the Zeta Function...... 73 § 1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum ...... 73 §2. Estimate of a Zeta Sum...... 82 §3. Estimate for the Zeta Function Close to the Line (1 = 1...... 86 §4. A Function-Theoretic Lemma ...... 87 §5. A New Boundary for the Zeros of the Zeta Function...... 88 §6. A New Remainder Term in the Prime Number Theorem...... 90 Exercises ...... 91

Chapter VII. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals. . . 94 § 1. The Simplest Density Theorem...... 94 §2. Prime Numbers in Short Intervals...... 98 Exercises ...... 100

Chapter VIII. Dirichlet L-Functions ...... 102 § 1. Characters and their Properties ...... 102 §2. Definition of L-Functions and their Simplest Properties...... 110 §3. The Functional Equation...... 113 §4. Non-trivial Zeros; Expansion of the Logarithmic Derivative as a Series in the Zeros...... 116 §5. Simplest Theorems on the Zeros ...... 117 Exercises ...... 119

Chapter IX. Prime Numbers in Arithmetic Progressions...... 122 §1. An Explicit Formula...... 122 §2. Theorems on the Boundary of the Zeros ...... 124 §3. The Prime Number Theorem for Arithmetic Progressions...... 135 Exercises...... 138

Chapter X. The Goldbach Conjecture ...... 141 §1. Auxiliary Statements...... 141 §2. The Circle Method for Goldbach's Problem...... 142 §3. Linear Trigonometric Sums with Prime Numbers...... 149 §4. An Effective Theorem...... 153 Exercises...... 158 Contents vii

Chapter XI. Waring's Problem ...... " 160 §l. The Circle Method for Waring's Problem...... 160 §2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem...... 171 §3. An Estimate for G(n)...... 174 Exercises ...... 177

Hints for the Solution of the Exercises...... 181 Table of Prime Numbers < 4070 and their Smallest Primitive Roots.. 217 Bibliography...... 219 Subject Index...... 221 Translator's Preface

This English translation of Karatsuba's Basic Analytic Number Theory follows closely the second Russian edition, published in Moscow in 1983. For the English edition, the author has considerably rewritten Chapter I, and has corrected various typographical and other minor errors throughout the the text.

August, 1991 Melvyn B. Nathanson Introduction to the English Edition

It gives me great pleasure that Springer-Verlag is publishing an English trans• lation of my book. In the Soviet Union, the primary purpose of this monograph was to introduce mathematicians to the basic results and methods of analytic number theory, but the book has also been increasingly used as a textbook by graduate students in many different fields of mathematics. I hope that the English edition will be used in the same ways. I express my deep gratitude to Professor Melvyn B. Nathanson for his excellent translation and for much assistance in correcting errors in the original text.

A.A. Karatsuba Introduction to the Second Russian Edition

Number theory is the study of the properties of the integers. Analytic number theory is that part of number theory in which, besides purely number theoretic arguments, the methods of mathematical analysis play an essential role. The purpose of this book is to provide an introduction to the central problems of analytic number theory. I have concentrated on the most important results, and have ignored excessively technical improvements. Consequently, I do not present the most recent refinements of the fundamental theorems. These latest results do not differ in principle from those that are presented in this book. The book focuses on four problems in analytic number theory: the problem of integer points in planar domains, the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions, Goldbach's problem, and Waring's problem. To solve these problems one must use the fundamental methods of analytic number theory: I.M. Vinogradov's method of trigonometric sums, the method of complex integration, and the circle method of G.H. Hardy, J.E. Littlewood, and S. Ramanujan. There are exercises at the end of each chapter. They are related to the contents of the chapter, and the reader should try to solve them. The problems either refine the theorems proved in the text, or lead to new ideas in number theory. The mathematical prerequisites for this book are undergraduate courses in number theory, mathematical analysis, and the theory of functions of a complex variable. This book should be read consecutively, since each chapter is connected with its predecessors. If a topic occurs several times in the book, then it will be explained in detail only at its first appearance. The reader should understand and be able to justify every mathematical argument, line by line. Only in this way will the study of the book be useful. The exercises in this book playa special role. Some of them are often very difficult, and can serve as topics for further research. The books [1]-[12] listed in the Bibliography discuss various mathematical topics related to this volume, and also contain historical and other references. Statements and formulas are numbered separately in each chapter. To refer to statements in other chapters, we note the chapter. This second edition is considerably different from the first. I have added new material to Chapter I on integer points, as well as hints for the solution of most of the exercises in the book. There are also new proofs for various theorems in Chapters III -- VII, X, and XI. Introduction to the Second Russian Edition xi

I received considerable assistance with this book from G.I. Arkhipov, S.M. Voronin, A.F. Lavrik, and V.N. Chubarikov. The manuscript was typed and edited by L.N. Abramochkina and R.I. Sorokina. I am very grateful to them all.

A.A. Karatsuba Notation

C, Co, C l' ... denote absolute positive constants, which are, in general, different in different theorems. For positive A, the notations B = O(A) and B ~ A mean that IBI ~ cA; the notation A::=::: B means that

c1A ~ B ~ c2 A.

C, c1 are arbitrarily small positive constants, and n, m, k, I, N denote natural numbers. Except in Chapter II, p, P1' . .. denote prime numbers. The Mobius function Jl(n) is defined by

1, if n = 1; Jl(n) = { 0, if n = p2 m; ( - 1)\ if n = P1 ... Pk For x > 0, the logarithm and the logarithmic integral are defined by

x du x du Inx = logx = J-; Lix = J-I - + Co, 1 U 2 nu where

The von Mangoldt function A(n) is defined by

A(n) = {loOgp if n = pk; if n#pk. The Euler function q>(k) is the number of natural numbers less than and relatively prime to k. Chebyshev's function t/J(x) and the prime counting function n(x) are defined by t/J(x) = L A(n); n(x) = I 1. n~x p5x

For/~k,(l,k)= 1,

t/J(x; k, I) = L A(n); n(x; k, I) = L 1. n=l(modk) p=l(modk) n~x p~x Notation xiii

The divisor function r( n) is the number of positive divisors of n; rk( n) denotes the number of solutions of the equation Xl X2 ... Xk = n in natural numbers Xl" .. ,Xk' Thus, r2(n) = r(n). The number of prime divisors of n is denoted by Q( n). For any real number tx, the integer part of tx, denoted by [tx], is the largest integer not exceeding tx; the fractional part of tx, denoted by { tx }, is tx - [tx]; and II tx II = min( {tx}, 1 - {tx}) is the distance from tx to the nearest integer. S = (1 + it denotes a_ complex number, where i2 = - 1, Re S = (1, Ims = t, S = (1 - it. In general,f is the complex conjugate of f Everywhere, In s = log s is the principal branch of the logarithm, and y is Euler's constant, defined by

y = lim (1 + -21 + . . . + ~ - In m ). m-++oo m The nontrivial zeros of the Riemann-zeta function and the Dirichlet L- series are enumerated in order of the increasing absolute value of their imaginary parts. A trigonometric sum is a finite sum of the form

L G(n)exp(2niF(n)), n';;P where G and F are real-valued functions of a natural number n, and