*(English)*Zbl 0756.11022

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. ${\Omega}$-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 $L$-functions; and finally, in the last chapter, a multidimensional ${\Omega}$-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.

##### MSC:

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |

11-02 | Research monographs (number theory) |

11M26 | Nonreal zeros of $\zeta \left(s\right)$ and $L(s,\chi )$; Riemann and other hypotheses |

11-01 | Textbooks (number theory) |

11M35 | Hurwitz and Lerch zeta functions |

11N05 | Distribution of primes |

11L15 | Weyl sums |

11M41 | Other Dirichlet series and zeta functions |