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

TL;DR

This paper develops a rigorous, efficient computational framework for the Epstein zeta function, including a high-performance C library, enabling advanced applications in quantum materials and multidimensional physics.

Contribution

It introduces a novel, compact representation and a superexponentially convergent algorithm for the Epstein zeta function, along with a high-performance implementation in EpsteinLib.

Findings

Achieved full precision computation across all parameters.

Enabled rapid evaluation of integrals involving the Epstein zeta function.

Applied to quantum dispersion relations and Casimir energies, revealing higher-order corrections.

Abstract

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.