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.

翻译：我们研究了利用欧拉-麦克劳林公式及新推导的一致渐近展开对Lerch超越函数$\Phi(z, s, a)$（其中$z, s, a \in \mathbb{C}$）进行任意精度数值评估的方法。对这些展开的详细分析附有严格的误差界。本文描述了针对参数与自变量取大值和小值的完整计算方案，并提供了实现高性能的算法细节。所描述的算法已在不同参数区间内经过广泛测试，并与当前最先进的代码进行了比较。基于本文所述算法的$\Phi(z, s, a)$开源实现可供使用。