Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib

We present a superexponentially convergent algorithm for computing the Epstein zeta function - a higher-dimensional generalization of the Riemann zeta function crucial for simulating quantum materials. We derive a compact and efficiently computable representation of the Epstein zeta function and examine its analytical properties across all arguments. To facilitate the computation of integrals involving the Epstein zeta function, we decompose it into a power-law singularity and a regularized Epstein zeta function, which is analytic in the first Brillouin zone. We present the first implementation of the Epstein zeta function and its regularization for arbitrary real arguments in EpsteinLib, a high-performance 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. We demonstrate the library's practical impact through applications to quantum dispersion relations in 3D spin materials with long-range interactions and Casimir energy calculations in multidimensional geometries, providing a powerful new tool for condensed matter physics and quantum materials research.