Spectral zeta functions for compact symmetric spaces of rank one

Abstract

We study the spectral zeta function ζM(s) associated with the spectrum of Laplacian acting on functions of a compact simply connected Riemannian symmetric space M of rank one and the spectral zeta function ζSnp, δ(s) associated with the spectrum of Laplacian acting on p-forms of the sphere Sn. We give the residues of ζM(s) and ζSnp(s) explicitly. For the odd dimensional sphere Sn, we show that ζSnp, δ(s) vanishes at negative integers.