Numerical methods and arbitrary-precision computation of the Lerch transcendent

TL;DR

This paper develops advanced numerical algorithms using the Euler-Maclaurin formula and asymptotic expansions for high-precision computation of the Lerch transcendent function, with rigorous error bounds and extensive testing.

Contribution

It introduces new uniform asymptotic expansions and a comprehensive computational scheme for the Lerch transcendent, enabling arbitrary-precision evaluation across diverse parameter regimes.

Findings

Algorithm achieves high accuracy and efficiency

Extensive testing confirms reliability across parameter ranges

Open-source implementation available for public use

Abstract

We examine the use of the Euler-Maclaurin formula and new derived uniform asymptotic expansions for the numerical evaluation of the Lerch transcendent $\Phi(z, s, a)$ for $z, s, a \in \mathbb{C}$ to arbitrary precision. A detailed analysis of these expansions is accompanied by rigorous error bounds. A complete scheme of computation for large and small values of the parameters and argument is described along with algorithmic details to achieve high performance. The described algorithm has been extensively tested in different regimes of the parameters and compared with current state-of-the-art codes. An open-source implementation of $\Phi(z, s, a)$ based on the algorithms described in this paper is available.