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An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

机译:一种有效的加速振荡序列收敛的算法,可用于计算多对数和Hurwitz zeta函数

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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.
机译:本文概述了一种提高一般振荡序列收敛速度的技术,然后将此序列加速算法应用于对数和Hurwitz zeta函数。因此,可以将其作为Borwein的“计算Riemann zeta函数的有效算法”给出的技术的扩展,扩展到更通用的系列。该算法为评估s的一般值和∣z 2 /(z-1)∣ < 4。通过使用复制公式和求逆公式,可将多对数的收敛范围扩展到整个复数z平面,因此,此处描述的算法允许针对所有复数s和z值评估多对数。另外,可以通过Euler-Maclaurin系列非常快速地评估Hurwitz zeta。多对数和Hurwitz zeta是相关的,因为可以使用对一个的两个评估来获得另一个的值。因此,这两种算法都可以用来评估任一函数。 Euler–Maclaurin系列显然是Hurwitz zeta性能的赢家,而Borwein算法在评估肾形区域的多对数方面表现优异。两种算法都优于简单的泰勒级数或直接求和。本文的主要而具体的结果是一种算法,该算法可以探索临界条带中的Hurwitz zeta,否则无法使用快速算法。包括对多对数单峰组的讨论。

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