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Modular functions and Dirichlet series in number theory. 2nd ed. (English) Zbl 0697.10023
Graduate Texts in Mathematics, 41. New York etc.: Springer-Verlag. x, 204 p. DM 98.00 (1990).

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”.

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 $\eta$ ($\tau \right)$. 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.

To conclude, this book will be high on the reviewer’s recommendation list for new students in the subject, as it has always been.

Reviewer: J.L.Hafner

##### MSC:
 11F03 Modular and automorphic functions 11-01 Textbooks (number theory) 11F67 Special values of automorphic $L$-series, etc 11F11 Holomorphic modular forms of integral weight 11G15 Complex multiplication and moduli of abelian varieties 11P81 Elementary theory of partitions 30B50 Dirichlet series, other series expansions, exponential series (one complex variable) 30F35 Fuchsian groups and automorphic functions 11M35 Hurwitz and Lerch zeta functions