*(English)*Zbl 0697.10023

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 $ ($\tau )$. 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.

##### 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 |