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Computational strategies for the Riemann zeta function. (English) Zbl 0972.11077

This fifty page paper provides a compendium of evaluation methods for the Riemann zeta-function. Formulas are given that range from historical attempts to recent found convergent series, with some curious oddities, old and new ones. The first section is devoted to provide a motivation for the search of efficient evaluation schemes. In the second, some properties of the Riemann zeta-function are listed, and the third is devoted to evaluations for general complex arguments. In the fourth section, rational zeta series are considered, while the fifth section focusses on integer arguments, specially on positive odd integer arguments, since the calculations for even ones can proceed through existing fast algorithms for computation of $\pi$ and its powers. Section six deals with alternative value-recycling schemes that can be successfully invoked for certain sets of arguments with integer differences. Section seven is devoted to evaluations of $\zeta$-values for integer arguments and in certain arithmetic progressions, with some detailed comments on the complexity issue. Finally, section eight contains some curiosities and open questions.

The paper, very valuable as a reference, concentrates primarily on practical computational issues, such issues depending on the domain of the argument, the speed of computation one wishes to achieve, and the incidence of what the author calls in the paper “value recycling”. It should be mentioned that, in some way, in the reviewer’s work: Ten physical applications of spectral zeta-functions. Lect. Notes Phys., New Ser. m35, Springer, Berlin (1995; Zbl 0855.00002), Commun. Math. Phys. 198, 83-95 (1998; Zbl 0932.11056) and J. Comput. Appl. Math. 118, 125-142 (2000; Zbl 1016.11034), there have been implemented similar ideas to other zeta functions, as the Hurwitz and Epstein zeta-functions, and generalizations thereof.

##### MSC:
 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11Y35 Analytic computations 11-02 Research monographs (number theory)