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 comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis.

Chapter 1

Elliptic functions

 1.1 Introduction

 1.2 Doubly periodic functions

 1.3 Fundamental pairs of periods

 1.4 Elliptic functions

 1.5 Construction of elliptic functions

 1.6 The Weierstrass function

 1.7 The Laurent expansion of near the origin

 1.8 Differential equation satisfied by

 1.9 The Eisenstein series and the invariants and

 1.10 The numbers el, e2,e3

 1.11 The discriminant

 1.12 Klein's modular function J()

 1.13 Invariance of J under unimodular transformations

 1.14 The Fourier expansions ofg2() and g3()

 1.15 The Fourier expansions of A() and J() Exercises for Chapter 1

Chapter 2 The Modular group and modular functions

 2.1 M6bius transformations

 2.2 The modular group F

 2.3 Fundamental regions

 2.4 Modular functions

 2.5 Special values of J

 2.6 Modular functions as rational functions of J

 2.7 Mapping properties of J

 2.8 Application to the inversion problem for Eisenstein series

 2.9 Application to Picard's theorem Exercises for Chapter 2

Chapter 3 The Dedekind eta function

 3.1 Introduction

 3.2 Siegers proof of Theorem 3.1

 3.3 Infinite product representation for A(r)

 3.4 The general functional equation for q(r)

 3.5 Iseki's transformation formula

 3.6 Deduction of Dedekind's functional equation from Iseki's formula

 3.7 Properties of Dedekind sums

 3.8 The reciprocity law for Dedekind sums

 3.9 Congruence properties of Dedekind sums

 3.10 The Eisenstein series G2(z) Exercises for Chapter

Chapter 4 Conyruences for the coefficients of the modular function

 4.1 Introduction

 4.2 The subgroup Fo(q)

 4.3 Fundamental region of Fo(p)

 4.4 Functions automorphic under the subgroup Fo(p)

 4.5 Construction of functions belonging to Fo(p)

 4.6 The behavior of ∫p under the generators of F

 4.7 The function φ(τ) = A(qτ)/A(τ)

 4.8 The univalent function φ(τ)

 4.9 Invariance of φ(ι) under transformations of Fo(q)

 4.10 The functionjp expressed as a polynomial in

Exercises for Chapter 4

Chapter 5

 Rademacher's series for the partition function

 5.1 Introduction

 5.2 The plan of the proof

 5.3 Dedekind's functional equation expressed in terms of F

 5.4 Farey fractions

 5.5 Ford circles

 5.6 Rademacher's path of integration

 5.7 Rademacher's convergent series for p(n)

Exercises for Chapter 5

Chapter 6

 Modular forms with multiplicative coefficients

 6.1 Introduction

 6.2 Modular forms of weight k

 6.3 The weight formula for zeros of an entire modular form

 6.4 Representation of entire forms in terms of G4 and G6

 6.5 The linear space Mk and the subspace Mk.o

 6.6 Classification of entire forms in terms of their zeros

 6.7 The Hecke operators Tn

 6.8 Transformations of order n

 6.9 Behavior of Tnfunder the modular group

 6.10 Multiplicative property of Hecke operators

 6.11 Eigenfunctions of Hecke operators

 6.12 Properties of simultaneous eigenforms

 6.13 Examples of normalized simultaneous eigenforms

 6.14 Remarks on existence of simultaneous eigenforms in M2k,o

 6.15 Estimates for the Fourier coefficients of entire forms

 6.16 Modular forms and Dirichlet series

Exercises for Chapter 6

Chapter 7

 Kronecker's theorem with applications

 7.1 Approximating real numbers by rational numbers

 7.2 Dirichlet's approximation theorem

 7.3 Liouville's approximation theorem

 7.4 Kronecker's approximation theorem: the one-dimensional case

 7.5 Extension of Kronecker's theorem to simultaneous approximation

 7.6 Applications to the Riemann zeta function

 7.7 Applications to periodic functions

Exercises for Chapter 7

Chapter 8

 General Dirichlet series and Bohr's equivalence theorem

 8.1 Introduction

 8.2 The half-plane of convergence of general Dirichlet series

 8.3 Bases for the sequence of exponents of a Dirichlet series

 8.4 Bohr matrices

 8.5 The Bohr function associated with a Dirichlet series

 8.6 The set of values taken by a Dirichlet seriesf(s) on a line σ=σ0

 8.7 Equivalence of general Dirichlet series

 8.8 Equivalence of ordinary Dirichlet series

 8.9 Equality of the sets Uι(σo) and Ug(σo) for equivalent Dirichlet series

 8.10 The set of values taken by a Dirichlet series in a neighborhood of the line σ=σ0

 8.11 Bohr's equivalence theorem

 8.12 Proof of Theorem 8.15

 8.13 Examples of equivalent Dirichlet series. Applications of Bohr's theorem to L-series

 8.14 Applications of Bohr's theorem to the Riemann zeta function Supplement to Chapter 3

Bibliography

Index of special symbols

Index