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

TL;DR

The paper develops a rigorous, high-precision framework for the Epstein zeta function $Z_{\Lambda,\nu}[\mathbf{x}|\mathbf{y}]$, delivering a compact Crandall-based representation that yields two rapidly convergent lattice sums and enables holomorphic continuation in $(\nu,\mathbf{x},\mathbf{y})$. It introduces a robust regularisation scheme $Z_{\Lambda,\nu}^{\mathrm{reg}}$ with a corresponding reg Crandall function to handle singularities, and provides a complete algorithm including preprocessing, special-case handling, controlled truncation, and stable evaluation of incomplete gamma functions. The authors present the first high-performance library EpsteinLib (C with Python and Mathematica bindings) that computes the Epstein zeta function and its regularisation with full precision up to dimension 10, and they validate accuracy and performance against known formulas. They demonstrate applications to quantum dispersion relations and Casimir energies, revealing higher-order corrections to asymptotic formulas and highlighting the practical impact for lattice sums in physics and materials science.

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.

Figures (3)

• Figure 1: The one dimensional Epstein zeta function in $\nu=1/2$ and $\Lambda =\mathds Z$ admits discontinuities in the (reciprocal) lattice, in particular in $x=y=0$ (a). The regularised Epstein zeta function in $y$ is analytic in the Brillouin zone $-1/2\le y\le 1/2$ and agrees with the Epstein zeta function in $y=0$ (b).
• Figure 2: Regions of different evaluation methods of the incomplete Gamma functions (a). Panel (b) offers a magnified view of the region close to the origin.
• Figure 3: Minimum of absolute and relative error of our implementation of the upper incomplete gamma function compared to the arbitrary precision result from Johansson2017arb. The dashed black box in (a) outlines the magnified region shown in (b).

Theorems & Definitions (30)

• definition 2.1: Lattices
• definition 2.2: Epstein zeta function
• theorem 2.3: Symmetries
• proof
• remark 2.4: Multi-atomic lattices
• definition 2.5: Fourier transform
• lemma 2.6: Poisson summation formula
• remark 2.7
• definition 2.8: Crandall functions
• lemma 2.9: Properties of Crandall functions