The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
type: Item
instance of: scholarly article
title: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
zbMATH Open document ID: 0756.11022
author: S. M. Voronin
author: Anatolii A. Karatsuba
published in: De Gruyter Expositions in Mathematics
publication date: 1993-01-23T00:00:00Z
review text: 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 acquaint readers with certain advances of the theory not covered by previous comprehensive treatises of the zeta- function such as the classic of \textit{E. C. Titchmarsh} (2nd ed., edited by \textit{D. R. Heath-Brown}) [The theory of the Riemann zeta-function. Oxford: Oxford University Press (1986; Zbl 0601.10026)], or the more recent monograph of \textit{A. Ivić} [The Riemann zeta-function. New York: 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 \(L\)-functi…
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11-01
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11L15
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 53026
zbMATH Keywords: approximate functional equations
zbMATH Keywords: density theorems
zbMATH Keywords: zeros on the critical line
zbMATH Keywords: distribution of nonzero values
zbMATH Keywords: universality
zbMATH Keywords: Riemann zeta- function
zbMATH Keywords: Vinogradov's method
zbMATH Keywords: \(\Omega\)-theorems
zbMATH Keywords: bibliography
zbMATH Keywords: Davenport-Heilbronn function
zbMATH Keywords: Hurwitz zeta-function
zbMATH Keywords: zeta-functions of quadratic forms
zbMATH Keywords: independence of \(L\)-functions
zbMATH Keywords: multidimensional \(\Omega\)-theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 53026