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Zbl 0697.10023
Apostol, Tom M.
Modular functions and Dirichlet series in number theory. 2nd ed. (English)
[B] Graduate Texts in Mathematics, 41. New York etc.: Springer-Verlag. x, 204 p. DM 98.00 (1990). ISBN 0-387-97127-0

The book under review is the second edition (for a review of the first (1976) see Zbl 0332.10017). The first was an excellent book that fully lived up to its intent as stated in the preface: ``It is hoped that these volumes will help the nonspecialist becomes 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''. \par This second addition differs very little from the first. The major difference is a very useful supplement to Chapter 3 on the Dedekind $\eta$-function. In that chapter, a proof of the functional equation for this function is given using Iseki's transformation formula, a very general relation involving log $\eta$ ($\tau)$. The alternate proof given in the supplement (suggested by B. Gordon) is much simpler, relying on the fact that the modular group has two generators, for which the functional equation is much easier to establish. \par To conclude, this book will be high on the reviewer's recommendation list for new students in the subject, as it has always been.
[J.L.Hafner]

MSC 2000:: *11F03 Modular and automorphic functions
11-01 Textbooks (number theory)
11F67 Special values of automorphic L-series, etc
11F11 Modular forms, one variable
11G15 Complex multiplication and moduli of abelian varieties
11P81 Elementary theory of partitions
30B50 Series expansions (one complex variable)
30F35 Fuchsian groups and automorphic functions
11M35 Other zeta functions

Keywords: modular functions; moduar forms; Dirichlet series; elliptic functions; Rademacher series; partition functions; Lehner congruences; Fourier coefficients; j-function; Dedekind $\eta$-function; transformation formula; modular group

Citations: Zbl 0332.10017

Cited in: Zbl 1201.11046 Zbl 0756.11011

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