Computation and properties of the Epstein zeta function: Application and numerical challenges
Date:
Session: Friday, August 23 Time: 09:30 - 09:50
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. In this work, we derive a compact and efficiently computable representation of the Epstein zeta function and examine its analytical properties across all arguments. We introduce a superexponentially convergent algorithm, including error bounds, for computing the Epstein zeta function in arbitrary dimensions. 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 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 Casimir energies in multidimensional geometries.
Speakers: Jonathan Busse, Ruben Gutendorf
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