The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. Beyond its numerous applications in pure mathematics, it has recently been identified as a key component in simulating exotic quantum materials. This work establishes the Epstein zeta function as a powerful tool in numerical analysis by rigorously investigating its analytical properties and enabling its efficient computation. Specifically, we derive a compact and computationally efficient representation of the Epstein zeta function and thoroughly examine its analytical properties across all arguments. Furthermore, we introduce a superexponentially convergent algorithm, complete with error bounds, for computing the Epstein zeta function in arbitrary dimensions. We also show that the Epstein zeta function can be decomposed into a power law singularity and an analytic function in the first Brillouin zone. This decomposition facilitates the rapid evaluation of integrals involving the Epstein zeta function and allows for efficient precomputations through interpolation techniques. We present the first high-performance implementation of the Epstein zeta function and its regularisation for arbitrary real arguments in EpsteinLib, a C library with Python and Mathematica bindings, and rigorously benchmark its precision and performance against known formulas, achieving full precision across the entire parameter range. Finally, we apply our library to the computation of quantum dispersion relations of three-dimensional spin materials with long-range interactions and Casimir energies in multidimensional geometries, uncovering higher-order corrections to known asymptotic formulas for the arising forces.

翻译：Epstein ζ函数将黎曼ζ函数推广至高维振荡格点求和。除在纯数学中的广泛应用外，该函数近期被确认为模拟奇异量子材料的关键组成部分。本文通过严格研究其解析性质并实现高效计算，确立了Epstein ζ函数在数值分析中的重要工具地位。具体而言，我们推导了Epstein ζ函数的紧凑且计算高效的表示形式，并系统研究了其所有参数下的解析性质。此外，我们提出了一种超指数收敛算法（附误差界），用于计算任意维度下的Epstein ζ函数。我们还证明了Epstein ζ函数在第一布里渊区内可分解为幂律奇点与解析函数之和。该分解显著提升了涉及Epstein ζ函数积分的计算效率，并支持通过插值技术进行高效预计算。我们在EpsteinLib（配备Python与Mathematica接口的C语言库）中首次实现了针对任意实参数的高性能Epstein ζ函数及其正则化计算，通过已知公式对精度与性能进行严格基准测试，实现了全参数范围内的完全精度。最后，我们将该库应用于三维长程相互作用自旋材料的量子色散关系计算，以及多维几何结构中卡西米尔能量的计算，从而揭示了相关作用力已知渐近公式的高阶修正项。