Modular Functions and Dirichlet Series in Number Theory

Tom M. Apostol (auth.)0387971270, 9780387971278

This volume is a sequel to the author’s Introduction to Analytic Number Theory (UTM 1976, 3rd Printing 1986). It presupposes an undergraduate background in number theory comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications. Among the major topics covered are Rademacher’s convergent series for the partition modular function, Lehner’s congruences for the Fourier coefficients of the modular function j, and Hecke’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. In addition to the correction of misprints, minor changes in the exercises and an updated bibliography, this new edition includes an alternative treatment of the transformation formula for the Dedekind eta function, which appears as a five-page supplement to Chapter 3.

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Table of contents :
Front Matter….Pages i-x
Elliptic functions….Pages 1-25
The modular group and modular functions….Pages 26-46
The Dedekind eta function….Pages 47-73
Congruences for the coefficients of the modular function j ….Pages 74-93
Rademacher’s series for the partition function….Pages 94-112
Modular forms with multiplicative coefficients….Pages 113-141
Kronecker’s theorem with applications….Pages 142-160
General Dirichlet series and Bohr’s equivalence theorem….Pages 161-189
Back Matter….Pages 190-198