We initiate the study of spectral zeta functions ζ X for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function ζ(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of ζ(s). We relate ζ Z to Euler's beta integral and show how to complete it giving the functional equation ξ Z (1 − s) = ξ Z (s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of ζ Z d (s) to the whole plane and identify the poles. From our aymptotics several known special values of ζ(s) are derived as well as its non-vanishing on the line Re(s) = 1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F 1 via an Euler-type integral formula due to Picard.

We prove an asymptotic formula for the determinant of the bundle Laplacian on discrete d-dimensional tori as the number of vertices tends to infinity. This determinant has a combinatorial interpretation in terms of cycle-rooted spanning forests. We also establish a relation (in the limit) between the spectral zeta function of a line bundle over a discrete torus, the spectral zeta function of the infinite graph Z d and the Epstein-Hurwitz zeta function. The latter can be viewed as the spectral zeta function of the twisted continuous torus which is the limit of the sequence of discrete tori.

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an asymptotic functional equation for these L-functions and the corresponding case of the Generalized Riemann Hypothesis. We also establish a relation between the positivity of such functions and the existence of real zeros in the critical strip of the classical Dirichlet L-functions with the same character.

No abstract

No abstract