Table Of ContentKaratsuba Basic Analytic Number Theory
Anatolij A. Karatsuba
Basic Analytic
Number Theory
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
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© Springer-Verlag Berlin Heidelberg 1993
Originally published by Springer-Verlag Berlin Heidelberg New York in 1993
Softcover reprinl of the hardcover 1s I edition 1993
Typesetting: Macmillan India Limited, Bangalore 560 025
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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
