Spectral zeta function on discrete tori and Epstein-Riemann hypothesis (Q2112792)

This authors first derive an asymptotic expansion of the spectral zeta function of the combinatorial Laplacian on a sequence of discrete tori which approximate the \(\alpha\)-dimensional torus for \(\alpha=2\). Then they prove that a certain conjecture on the asymptotics gives an equivalence formulation of the Epstein-Riemann Hypothesis for \(\alpha=2\), if they replace the standard discrete Laplacian with the 9-point star discrete Laplacian. Here the Epstein-Riemann Hypothesis asserts that the spectral zeta function of the torus, which is the Epstein zeta function for the identity matrix of size \(\alpha\), satisfies the analog of the Riemann Hypothesis having all nontrivial zeros in the critical strip \(0<\mathrm{Re}(s)<\alpha/2\) on the line \(\mathrm{Re}(s)=\alpha/4\). These results generalize the theorems of Friedli and Karlsson who obtained an asymptotic formula for \(\alpha=1\) and proved the equivalence between a certain conjecture on it and the Riemann Hypothesis.