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Zbl 0756.11022
Karatsuba, A.A.; Voronin, S.M.
The Riemann zeta-function. Transl. from the Russian by Neal Koblitz. (English)
[B] De Gruyter Expositions in Mathematics. 5. Berlin etc.: W. de Gruyter. xii, 396 p. (1992). ISBN 3-11-013170-6/hbk

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 {\it E. C. Titchmarsh} (2nd ed., edited by {\it D. R. Heath-Brown}) [``The theory of the Riemann zeta-function'' (Oxford University Press 1986; Zbl 0601.10026)], or the more recent monograph of {\it 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.\par 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.\par 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.
[M.Jutila (Turku)]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function
11-02 Research monographs (number theory)
11M26 Nonreal zeros of zeta(s) and L(s,chi)
11-01 Textbooks (number theory)
11M35 Other zeta functions
11N05 Distribution of primes
11L15 Weyl sums
11M41 Other Dirichlet series and zeta functions

Keywords: approximate functional equations; density theorems; zeros on the critical line; distribution of nonzero values; universality; Riemann zeta- function; Vinogradov's method; $\Omega$-theorems; bibliography; Davenport-Heilbronn function; Hurwitz zeta-function; zeta-functions of quadratic forms; independence of $L$-functions; multidimensional $\Omega$-theorem

Citations: Zbl 0601.10026; Zbl 0556.10026

Cited in: Zbl 1073.11005 Zbl 1040.11080 Zbl 0982.11050 Zbl 0836.11029

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