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Series associated with the zeta and related functions. (English) Zbl 1014.33001

Dordrecht: Kluwer Academic Publishers. ix, 388 p. (2001).
A large amount of literature exists concerning the Riemann zeta and allied functions, much of it in widely scattered journals. Consequently, any systematic compendium of known results is to be welcomed. No single volume can do justice to all facets of this field, and the authors have confined their efforts to collecting and organizing families of infinite series, particularly those that can be evaluated in closed form. As expected, the choice of topics is dictated in large part by their own research interests.
Chapter 1 (Introduction and preliminaries) helps make the book more or less self-contained for readers with knowledge of complex analysis by including background material on topics such as gamma, beta, polygamma, and multiple gamma functions, hypergeometric functions, Stirling numbers of the first and second kind, as well as Bernoulli and Euler numbers and polynomials.
Chapter 2 (The zeta and related functions) introduces definitions and basic properties of the Riemann zeta function, the Hurwitz and Lerch zeta functions, polylogarithms, and multiple zeta functions. The point of departure is the multiple Hurwitz function \(\zeta_n(s,a)\) defined by the multiple series \(\sum (a+k_1+\cdots+ k_n)^{-s}\) extended over nonnegative integers \(k_1,\dots, k_n\), where \(\operatorname{Re}(s)>n\), and \(a\) is not zero or a negative integer. It can be extended as a meromorphic function to the entire \(s\) plane with simple poles at the integers \(s=1,2,\dots, n\). The special case \(n=1\) is the usual Hurwitz zeta function \(\zeta(s,a)\) which, in turn, includes the Riemann zeta function \(\zeta(s)\) as the special case \(a=1\).
Chapter 3 (Series involving zeta functions) is a lengthy chapter that describes methods and techniques used to obtain closed-form evaluation of series involving zeta functions. Section 3.4, which is ninety pages long, contains a list of more than 700 formulas, half of which are obtained by starting from the closed form evaluation of the power series \(\sum_{k=2}^\infty \zeta(k) t^{k-1}\) and variations of it obtained by differentiation, integration, addition and subtraction, and then specializing the value of \(t\). Later in the chapter recursion formulas involving the zeta function are given for higher-order derivatives of the gamma function evaluated at 1 and at \(\frac 12\). Explicit formulas for \(\Gamma^{(n)}(1)\) and \(\Gamma^{(n)} (\frac 12)\) are listed for \(n= 1,\dots, 10\).
In Chapter 4 (Evaluations and series representations) the primary emphasis is closed-form evaluations and rapidly convergent series representations of \(\zeta(n)\) for integer \(n\geq 2\).
Chapter 5 (Determinants of the Laplacian) gives applications of results in earlier chapters to the computation of determinants of the Laplacians on Riemannian manifolds of constant curvature, particularly the evaluation of the functional determinant for the \(n\)-dimensional sphere \(\mathbb{S}^n\) with the standard metric.
Chapter 6 (Miscellaneous results), the last and shortest chapter (fewer than 20 pages), evaluates Bernoulli and Euler polynomials and several trigonometric series for rational arguments, as well as some integrals associated with the Euler-Maclaurin summation formula.
This is followed by a 26-page bibliography, an author index, and a subject index. The formulas are easy to read, and the pages are clean and uncluttered. The end of each chapter contains so-called problems that presumably can be solved using material in that chapter. The level of difficulty of these problems varies considerably. Some can be solved by specializng the value of a parameter in a general formula. Others represent major pieces of research taken (with appropriate references) from recent publications.
Since this is really a reference work rather than a textbook, the reviewer feels that the problem should be considered as miscellaneous results akin to those in Chapter 6. In a revision, the authors may wish to add material on closed-form formulas for higher derivatives of the Riemann zeta function at 0, extending Eq. 2.2(20) for \(\zeta'(0)\), especially because they are related to the number \(\Gamma^{(n)}(1)\) listed in Section 3.6.
Connections with number theory are not included, except for a brief remark on p. 97 to the effect that \(\zeta(s)\) plays a central role in the applications of complex analysis to number theory because of Euler’s representation of \(\zeta(s)\) as an infinite product extended over all primes.
Any reviewer would like to see a useful book made even more useful, and the following criticisms should be considered in this light. First, the book needs a separate index of notations and special symbols. This reviewer could not find an explicit definition of the double gamma function, which is denoted both by \(\Gamma_2\) and by \(G= 1/\Gamma_2\), and which is also referred to as the Barnes \(G\)-function. Equations in each section are numbered separately; when referring to an equation in another section, a number like 3.4(18) is used to represent Equation (18) in Section 3.4. This numbering system makes it hard for the reader to navigate within a given chapter. For example, Section 3.4 is ninety pages long and one has to consult the Table of Contents to find where Section 3.4 begins. It would have helped to incorporate section numbers in the running heads as was done, for example, in “Higher transcendental functions” by Erdélyi et al.
It is virtually impossible to publish a compendium such as this that is free from errors, and the authors do not explain what precautions were used to insure accuracy. In spot-checking some formulas familiar to this reviewer, the following errors were found: On p. 19 in Equation (48), \(\gamma\) should be \(-\gamma\); on p. 103 in Equation (45), \(s\) should not appear in the denominator of the second formula; in Equation (47) the first plus sign should be a minus sign (the same error appears in “Higher transcendental functions”); and in Equation (51) the factor 2 should not appear in the denominator. This prompted further spot-checking, but no other errors were found by this reviewer.
In summary, this is an impressive labor of love, and both the authors and the publisher should be commended for sharing it with the mathematical community.

MSC:

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33B15 Gamma, beta and polygamma functions
33Cxx Hypergeometric functions
11M41 Other Dirichlet series and zeta functions
65B15 Euler-Maclaurin formula in numerical analysis
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11Y60 Evaluation of number-theoretic constants
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