Spectral zeta function on discrete tori and Epstein-Riemann conjecture

Autor:	Meiners, Alexander, Vertman, Boris
Rok vydání:	2022
Předmět:	Mathematics - Spectral Theory Mathematics - Differential Geometry 52C05 58J52 65B15
Druh dokumentu:	Working Paper
Popis:	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. Comment: 37 pages, 2 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2202.02420 Zobrazit plný text záznamu View this record from Arxiv