Spectral zeta function on discrete tori and Epstein-Riemann conjecture

Abstract

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the α-dimensional torus. In the special case α = 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 α = 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 9point star discrete Laplacian.