The zeta function for circular graphs

Abstract. We look at entire functions given as the zeta function λ>0 λ −s, where λ are the positive eigenvalues of the Laplacian of the discrete circular graph Cn. We prove that the roots converge for n → ∞ to the line σ = Re(s) = 1/2 in the sense that for every compact subset K in the complement of that line, there is a nK such that for n> nK, no root of the zeta function is in K. To prove the result we actually look at the Dirac zeta function ζn(s) = λ>0 λ −s where λ are the positive eigenvalues of the Dirac operator of the circular graph. In the case of circular graphs, the Laplace zeta function is ζn(2s). 1. Extended summary The zeta function ζG(s) for a finite simple graph G is the entire function λ>0 λ −s defined by the positive eigenvalues λ of the Dirac operator D = d+ d∗, the square root of the Hodge Laplacian L = dd∗+d∗d on discrete differential forms. We study it in the case of circular graphs G = Cn, where ζn(2s) agrees with the zeta function∑ λ>0,λ∈σ(L) λ−s of the classical scalar Laplacian lik