Spectral zeta function on discrete tori and Epstein-Riemann conjecture

TL;DR

This paper extends the asymptotic analysis of spectral zeta functions on discrete tori to higher dimensions, linking these results to the Epstein-Riemann conjecture through a generalized Laplacian.

Contribution

It proves that similar asymptotic expansions hold for spectral zeta functions in higher dimensions, providing a new formulation of the Epstein-Riemann conjecture.

Findings

Asymptotic expansion holds for m=2

Method applies to higher dimensions

Formulation of Epstein-Riemann conjecture via generalized Laplacian

Abstract

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for m=2. Similar argument applies to higher dimensions as well. A conjecture on this asymptotics gives an equivalent formulation of the Epstein-Riemann conjecture, if we replace the standard discrete Laplacian with the $9$-point star discrete Laplacian.