De Gruyter Expositions in Mathematics. 5. Berlin etc.: W. de Gruyter. xii, 396 p. (1992).
The aims of this book are twofold: first, to serve as an introduction to the theory of the Riemann zeta-function with its number theoretic applications, and second, to aquaint readers with certain advances of the theory not covered by previous comprehensive treatises of the zeta- function such as the classic of E. C. Titchmarsh (2nd ed., edited by D. R. Heath-Brown) [“The theory of the Riemann zeta-function” (Oxford University Press 1986; Zbl 0601.10026)], or the more recent monograph of A. Ivić [“The Riemann zeta-function” (Wiley 1985; Zbl 0556.10026)]. Thus the choice of the more advanced material is intentionally selective; for instance, there is no discussion of mean value problems and results for the zeta-function.
The headings of the chapters give an idea of the contents: I. The definition and simplest properties of the Riemann zeta-function, II. The Riemann zeta-function as a generating function in number theory, III. Approximate functional equations, IV. Vinogradov’s method in the theory of the Riemann zeta-function, V. Density theorems, VI. Zeros of the zeta- function on the critical line, VII. Distribution of nonzero values of the Riemann zeta-function, VIII. -theorems. In addition, there is an extensive appendix containing various auxiliary results, and a bibliography of 172 references.
Most of the new or less standard material, mainly originating from the research of the authors, can be found in the last four chapters. To give a few examples, results of the Selberg type on zeros lying on or near the critical line are given as “local” versions; the distribution of the zeros of the Davenport-Heilbronn function, the Hurwitz zeta-function and zeta-functions of quadratic forms (all having a functional equation but not an Euler product) is discussed in detail; further, there are theorems about the “universality” of the zeta-function and allied functions, as well as about the independence of -functions; and finally, in the last chapter, a multidimensional -theorem is proved in addition to more standard results of this kind. These examples also indicate that the scope of this well-written book is by no means restricted to the Riemann zeta-function. It spans successfully from elementary theory to topics of recent and current research.