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 kidney-shaped 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 z-plane, 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 kidney-shaped 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. L. Vepštas (B)

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

“…Finally, it would be interesting to compare the efficiency of the Euler-Maclaurin formula with other approaches to evaluating the Hurwitz zeta function such as the algorithms of Borwein [8], Vepštas [38] and Coffey [10].…”

The Lerch Zeta function III. Polylogarithms and special values

“…Very recently, Costin and Garoufalidis [16] obtained a multivalued analytic continuation for the function ζ (x, s), calling it the "fractional polylogarithm" and denoting it Li α (x) = ∞ n=1 x n n α in variables (α, x) on a cover of C × (P 1 (C) {0, 1, ∞}); such a continuation appears here as a special case of Theorem 3.6. Vepstas [72] also obtained results applicable to analytic continuation of the fractional polylogarithm.…”

Zeros of the dilogarithm

“…Clearly Li 1 (0) = 0 gives the only zero on the principal branch and Li 1 (z) is non-zero on every other branch. Going in the other direction, if we let Li 3 (z) denote the trilogarithm on its principal branch, it may be shown (see [Vep08,p. 246]) that on any branch it has the form…”