Spectral zeta functions for compact symmetric spaces of rank one (Q5929127)

The paper studies the spectral zeta functions \(\xi_M(s)\) and \(\xi_M^{p,\delta}(s)\) associated with the spectrum of the Laplacian \(\Delta_M\) acting on functions and on \(p\)-forms on \(M\), correspondingly. Here \(M\) is a compact simply connected Riemannian symmetric space of rank one. The functions \(\xi_M(s)\) and \(\xi_M^{p,\delta}(s)\) are meromorphic. The author describes explicitly the residues of \(\xi_M(s)\) (where \(M\) is as above) and \(\xi_{S^n}^{p,\delta}(s)\). Besides, it is shown in the paper that for \(n\) odd the function \(\xi_{S^n}^{p,\delta}(s)\) vanishes at negative integers.