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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 \(\eta\) (\(\tau)\). 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.

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.

Reviewer: J.L.Hafner

### MSC:

11F03 | Modular and automorphic functions |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F11 | Holomorphic modular forms of integral weight |

11G15 | Complex multiplication and moduli of abelian varieties |

11P81 | Elementary theory of partitions |

30B50 | Dirichlet series, exponential series and other series in one complex variable |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

11M35 | Hurwitz and Lerch 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
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\textit{T. M. Apostol}, Modular functions and Dirichlet series in number theory. 2nd ed. New York etc.: Springer-Verlag (1990; Zbl 0697.10023)

### Digital Library of Mathematical Functions:

§17.2(i) 𝑞-Calculus ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions§23.10(iv) Homogeneity ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.15(i) General Modular Functions ‣ §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.17(ii) Power and Laurent Series ‣ §23.17 Elementary Properties ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.18 Modular Transformations ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Dedekind’s Eta Function ‣ §23.18 Modular Transformations ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.19 Interrelations ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Rhombic Lattice ‣ §23.20(i) Conformal Mappings ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(v) Modular Functions and Number Theory ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

§27.14(iii) Asymptotic Formulas ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory

§27.14(iv) Relation to Modular Functions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory

§27.14(vi) Ramanujan’s Tau Function ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory

§27.7 Lambert Series as Generating Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory

Chapter 27 Functions of Number Theory