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The Riemann zeta-function. The theory of the Riemann zeta-function with applications. (English) Zbl 0556.10026
A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XVI, 517 p. £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 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 H. M. Edwards [Riemann’s zeta function (1974; Zbl 0315.10035)], which is principally a historical account.

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.

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 ${p}_{n+1}-{p}_{n}\ll {p}_{n}^{13/23}$, 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.

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 $m\left(\sigma \right)$, for different ranges of $\sigma$; one of these is $m\left(\sigma \right)\ge 12408/\left(4357-4890\sigma \right)$ 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($\sigma \right)$ 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 $A{B}^{1-r}\ll |{f}^{\left(r\right)}\left(x\right)|\ll A{B}^{1-r}$ in the discussion of exponent pairs, and believes this new, weaker, hypothesis may be inadequate.)

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).]

Reviewer: D.R.Heath-Brown

##### MSC:
 11-02 Research monographs (number theory) 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11N37 Asymptotic results on arithmetic functions 11L40 Estimates on character sums