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Zbl 0556.10026
Ivić, Aleksandar
The Riemann zeta-function. The theory of the Riemann zeta-function with applications. (English)
[B] A Wiley-Interscience Publication. New York etc.: John Wiley \& Sons. XVI, 517 p. \sterling 57.80 (1985).

The purpose of this book is to present an up to date account of the theory of the Riemann zeta-function, together with some of its applications. The well known book by {\it E. C. Titchmarsh} [The theory of the Riemann zeta-function (1951; Zbl 0042.07901)] contains material only from 1950 or earlier, so that there is much recent and not-so-recent work to cover. The only other major modern book on the subject is that by {\it H. M. Edwards} [Riemann's zeta function (1974; Zbl 0315.10035)], which is principally a historical account. \par Naturally not all the developments of the last 35 years could be included in this volume, but the following list of contents gives a fair indication of the scope of the book: Elementary theory, Exponential integrals and exponential sums, The Voronoi summation formula, The approximate functional equations, The fourth power moment, The zero-free region, Mean value estimates over short intervals, Higher power moments, Omega results, Zeros on the critical line, Zero-density estimates, The distribution of primes, The Dirichlet divisor problem, Various other divisor problems, Atkinson's formula for the mean square. \par Of particular interest are the accounts of Voronoi's summation formula and its applications, and of Atkinson's formula for the mean square of $\zeta$ (s). Some of the material is, however, considerably compressed. The result of Iwaniec and Jutila, that $p\sb{n+1}-p\sb n\ll p\sb n\sp{13/23}$, is dealt with in just 6 pages, for example. As is so often the case, the author's end of chapter notes form perhaps the best part of the book, describing a multitude of additional results, with comments thereon. There is a bibliography of some 350 references. \par Much of the work shows the author's own influence and interests. In particular the chapters on higher power moments, zero-density theorems and divisor problems contain many of his results, which are currently the best known. These depend heavily on the estimation of exponential sums by the method of exponent pairs, which fact is reflected in the flavour of the results. Thus, for example, Theorem 8.4 gives 8 different lower bounds for a certain quantity $m(\sigma)$, for different ranges of $\sigma$; one of these is $m(\sigma)\ge 12408/(4357-4890 \sigma)$ when $3/4\le \sigma \le 5/6$. The reviewer feels that such a result is essentially useless - it is not even obvious whether m($\sigma)$ is positive or not. The level of exposition is generally good, but in one or two places the logic becomes confused. (For example, the reviewer could not follow the treatment of the condition $AB\sp{1-r}\ll \vert f\sp{(r)}(x)\vert \ll AB\sp{1-r}$ in the discussion of exponent pairs, and believes this new, weaker, hypothesis may be inadequate.) \par Such criticisms aside the author has clearly provided a very significant and much needed service by giving a unified account of so much important work. [See also the preliminary version, Topics in recent zeta function theory (Publ. Math. Orsay 83.06) (1983; Zbl 0524.10032).]
[D.R.Heath-Brown]

MSC 2000:: *11-02 Research monographs (number theory)
11M06 Riemannian zeta-function and Dirichlet L-function
11N37 Asymptotic results on arithmetic functions
11L40 Estimates on character sums

Keywords: Riemann zeta-function; Elementary theory; Exponential integrals; exponential sums; approximate functional equations; fourth power moment; zero-free region; Mean value estimates over short intervals; Omega results; Zeros on the critical line; Zero-density estimates; distribution of primes; Dirichlet divisor problem; Voronoi's summation formula; Atkinson's formula; mean square; bibliography; higher power moments; zero-density theorems; divisor problems; estimation of exponential sums; exponent pairs

Citations: Zbl 0042.07901; Zbl 0315.10035; Zbl 0524.10032

Cited in: Zbl pre06101583 Zbl 1226.11086 Zbl 1050.11075 Zbl 1036.11045 Zbl 1034.11046 Zbl 1028.11050 Zbl 1022.11047 Zbl 1006.11048 Zbl 0977.11042 Zbl 0965.11040 Zbl 0948.11032 Zbl 0909.11001 Zbl 0891.11040 Zbl 0836.11029 Zbl 0793.11022 Zbl 0798.11036 Zbl 0823.11051 Zbl 0763.11039 Zbl 0756.11022 Zbl 0742.11045 Zbl 0659.10053 Zbl 0655.10034 Zbl 0684.10035 Zbl 0678.10027 Zbl 0619.10041 Zbl 0665.10028 Zbl 0654.10041 Zbl 0652.10033 Zbl 0644.10031 Zbl 0627.10027 Zbl 0601.10026 Zbl 0589.10053 Zbl 0573.10035 Zbl 0573.10027 Zbl 0545.10026

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