Singular Euler-Maclaurin expansion on multidimensional lattices

TL;DR

This paper generalizes the Euler-Maclaurin expansion to multidimensional lattices with singular functions, enabling precise analysis of microscopic effects on macroscopic properties, with applications in physics and efficient computational methods.

Contribution

The authors develop a multidimensional singular Euler-Maclaurin expansion, extending previous one-dimensional work, and connect it to number theory for efficient computation of lattice sums.

Findings

Effective numerical computation of singular lattice sums in 2D.

Extension of Euler-Maclaurin to higher dimensions with singularities.

Implementation available in Mathematica for practical use.

Abstract

We extend the classical Euler-Maclaurin expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for the precise quantification of the effect of microscopic discreteness on macroscopic properties of a system. First, the Euler-Maclaurin summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler-Maclaurin (SEM) expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of high relevance in solid state and quantum physics. An implementation in Mathematica is provided online along with this article.