An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
Abstract
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein?s ?An efficient algorithm for computing the Riemann zeta function? by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li s (z) for general values of complex s and a kidneyshaped region of complex z values given by ?z 2/(z?1)?<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex zplane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler?Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler?Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidneyshaped region. Both algorithms are superior to the simple Taylor?s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.
 Publication:

Numerical Algorithms
 Pub Date:
 March 2008
 DOI:
 10.1007/s1107500791538
 arXiv:
 arXiv:math/0702243
 Bibcode:
 2008NuAlg..47..211V
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Numerical Analysis;
 Mathematics  Number Theory;
 65B10 (primary);
 11M35;
 11Y35;
 33F05;
 68W25 (secondary)
 EPrint:
 37 pages, 6 graphs, 14 fullcolor phase plots. v3: Added discussion of a fast Hurwitz algorithm