Apostol T.M. Modular..

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41
Graduate Texts in Mathematics
Editorial Board
J.H. Ewing F.W. Gehring P.R. Halmos
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contmuttd after lndel
Tom M. Apostol
Modular Functions
and Dirichlet Series
in Number Theory
Second Edition
With 25 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo Hong Kong
Tom M. Apostol
Department of Mathematics
California Institute of Technology
Pasadena, CA 91125
USA
Editorial Board
J.H. Ewing F. W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Indiana University University of Michigan Santa Clara University
Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
AMS Subject Classifications
IOA20, 10A45, IOD45, 10H05, !OHIO, 10120, 30Al6
Library of Congress Cataloging-in-Publication Data
Apostol, Tom M.
Modular functions and Dirichlet series in number theory/Tom M. Apostol.-2nd ed.
p. cm.-(Graduate texts in mathematics; 41)
Includes bibliographical references.
ISBN 0-387-97127-0 (alk. paper)
I. Number theory. 2. Functions, Elliptic. 3. Functions, Modular. 4. Series,
Dirichlet. I. Title. II. Series.
QA241.A62 1990
512'.7---dc20 89-21760
Printed on acid-free paper.
© 1976, 1990 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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9 8 7 6 5 4 3 2 I
ISBN 0-387-97127-0 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540-97127-0 Springer-Verlag Berlin Heidelberg New York
Preface
This is the second volume of a 2-volume textbook* which evolved from a
course (Mathematics 160) offered at the California Institute of Technology
during the last 25 years.
The second volume presupposes a background in number theory com
parable to that provided in the first volume, together with a knowledge of
the basic concepts of complex analysis.
Most of the present volume is devoted to elliptic functions and modular
functions with some of their number-theoretic applications. Among the
major topics treated are Rademacher's convergent series for the partition
function, Lehner's congruences for the Fourier coefficients of the modular
functionj('r), and Heeke's theory of entire forms with multiplicative Fourier
coefficients. The last chapter gives an account of Bohr's theory of equivalence
of general Dirichlet series.
Both volumes of this work emphasize classical aspects of a subject which
in recent years has undergone a great deal of modern development. It is
hoped that these volumes will help the nonspecialist become acquainted
with an important and fascinating part of mathematics and, at the same
time, will provide some of the background that belongs to the repertory of
every specialist in the field.
This volume, like the first, is dedicated to the students who have taken
this course and have gone on to make notable contributions to number
theory and other parts of mathematics.
T.M.A.
January, 1976
*The first volume is in the Springer-Verlag series Undergraduate Texts in Mathematics under
the title Introduction to Analytic Number Theory.
v
Preface to the Second Edition
The major change is an alternate treatment of the transformation formula for
the Dedekind eta function, which appears in a five-page supplement to Chap
ter 3, inserted at the end of the book (just before the Bibliography). Other
wise, the second edition is almost identical to the first. Misprints have been
repaired, there are minor changes in the Exercises, and the Bibliography has
been updated.
T.M.A.
July, 1989
Contents
Chapter 1
Elliptic functions
1.1 Introduction 1
1.2 Doubly periodic functions 1
1.3 Fundamental pairs of periods 2
1.4 Elliptic functions 4
1.5 Construction of elliptic functions 6
1.6 The Weierstrass p function 9
1. 7 The Laurent expansion of p near the origin 11
1.8 Differential equation satisfied by f.J 11
1.9 The Eisenstein series and the invariants 9 and 9 12
2 3
1.10 The numbers e1, e2, e3 13
1.11 The discriminant A 14
1.12 Klein's modular function J(r) 15
1.13 Invariance of J under unimodular transformations 16
1.14 The Fourier expansions of 9z(r) and 9 (-r:) 18
3
1.15 The Fourier expansions of A(r) and J(r) 20
Exercises for Chapter 1 23
Chapter 2
The Modular group and modular functions
2.1 Mobius transformations 26
2.2 The modular group r 28
2.3 Fundamental regions 30
2.4 Modular functions 34
vii
2.5 Special values of J 39
2.6 Modular functions as rational functions of J 40
2. 7 Mapping properties of J 40
2.8 Application to the inversion problem for Eisenstein series 42
2.9 Application to Picard's theorem 43
Exercises for Chapter 2 44
Chapter 3
The Dedekind eta function
3.1 Introduction 47
3.2 Siegel's proof of Theorem 3.1 48
3.3 Infinite product representation for Ll( r) 50
3.4 The general functional equation for ry(r) 51
3.5 Iseki's transformation formula 53
3.6 Deduction of Dedekind's functional equation from Iseki's
formula 58
3.7 Properties of Dedekind sums 61
3.8 The reciprocity law for Dedekind sums 62
3.9 Congruence properties of Dedekind sums 64
3.10 The Eisenstein series G (r) 69
2
Exercises for Chapter 3 70
Chapter 4
Congruences for the coefficients of the modular function j
4.1 Introduction 74
4.2 The subgroup r (q) 75
0
4.3 Fundamental region of r (p) 76
0
4.4 Functions automorphic under the subgroup r (p) 78
0
4.5 Construction of functions belonging tor (p) 80
0
4.6 The behavior of fP under the generators of r 83
4.7 The function <p(r) = Ll(qr)/Ll(r) 84
4.8 The univalent function <l>(r) 86
4.9 Inv ariance of <I>( r) under transformations of r (q) 87
0
4.10 The functionjP expressed as a polynomial in <I> 88
Exercises for Chapter 4 91
Chapter 5
Rademacher's series for the partition function
5.1 Introduction 94
5.2 The plan of the proof 95
5.3 Dedekind's functional equation expressed in terms ofF 96
5.4 Farey fractions 97
viii
5.5 Ford circles 99
5.6 Rademacher's path of integration 102
5. 7 Rademacher's convergent series for p(n) 104
Exercises for Chapter 5 110
Chapter 6
Modular forms with multiplicative coefficients
6.1 Introduction 113
6.2 Modular forms of weight k 114
6.3 The weight formula for zeros of an entire modular form 115
6.4 Representation of entire forms in terms of G and G 117
4 6
6.5 The linear space Mk and the subspace Mk, 118
0
6.6 Classification of entire forms in terms of their zeros 119
6. 7 The Heeke operators Tn 120
6.8 Transformations of order n 122
6.9 Behavior of Tnfunder the modular group 125
6.10 Multiplicative property of Heeke operators 126
6.11 Eigenfunctions of Heeke operators 129
6.12 Properties of simultaneous eigenforms 130
6.13 Examples of normalized simultaneous eigenforms 131
6.14 Remarks on existence of simultaneous eigenforms in M lk, 0 133
6.15 Estimates for the Fourier coefficients of entire forms 134
6.16 Modular forms and Dirichlet series 136
Exercises for Chapter 6 138
Chapter 7
Kronecker's theorem with applications
7.1 Approximating real numbers by rational numbers 142
7.2 Dirichlet's approximation theorem 143
7.3 Liouville's approximation theorem 146
7.4 Kronecker's approximation theorem: the one-dimensional
case 148
7.5 Extension of Kronecker's theorem to simultaneous
approximation 149
7.6 Applications to the Riemann zeta function 155
7.7 Applications to periodic functions 157
Exercises for Chapter 7 159
Chapter 8
General Dirichlet series and Bohr's equivalence theorem
8.1 Introduction 161
8.2 The half-plane of convergence of general Dirichlet series 161
8.3 Bases for the sequence of exponents of a Dirichlet series 166
IX
8.4 Bohr matrices 167
8.5 The Bohr function associated with a Dirichlet series 168
8.6 The set of values taken by a Dirichlet seriesf(s) on a line
u = u 170
0
8. 7 Equivalence of general Dirichlet series 173
8.8 Equivalence of ordinary Dirichlet series 174
8.9 Equality of the sets UtCu and Ug(u for equivalent
0) 0)
Dirichlet series 176
8.10 The set of values taken by a Dirichlet series in a neighborhood
of the line u = u 176
0
8.11 Bohr's equivalence theorem 178
8.12 Proof of Theorem 8.15 179
8. 13 Examples of equivalent Dirichlet series. Applications of Bohr's
theorem to £-series 184
8.14 Applications of Bohr's theorem to the Riemann zeta function 184
Exercises for Chapter 8 187
Supplement to Chapter 3 190
Bibliography 196
Index of special symbols 199
Index 201
X