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：我们研究使用Euler-Maclaurin公式和新衍生的统一无症状扩展公式,对Lerch Explent $\Phi(z, s, a) $z, s, a) $ $ $, a $ $, a mathbb{C} 美元进行任意精确的数值评估。对这些扩展的详细分析附有严格的误差界限。对参数和参数和参数的大小值的完整计算方法以及实现高性能的算法细节进行了描述。描述的算法已经在不同参数体系中进行了广泛测试,并与目前的最新代码进行了比较。基于本文所描述的算法的 $\ Phi(z, s, a) 的公开源实施。
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