Special values of the spectral zeta functions for locally symmetric Riemannian manifolds

Abstract

In this paper, we establish the formulas expressing the special values of the spectral zeta function ${\zeta }_{\mathrm{\Delta }}\left(n\right)$ of the Laplacian $\mathrm{\Delta }$ on some locally symmetric Riemannian manifold $\mathrm{\Gamma }\backslash G/K$ in terms of the coefficients of the Laurent expansion of the corresponding Selberg zeta function. As an application, we give a numerical estimation of the first eigenvalue of $\mathrm{\Delta }$ by computing the values ${\zeta }_{\mathrm{\Delta }}\left(n\right)$ numerically, when $\mathrm{\Gamma }\backslash G/K$ is a Riemann surface with $\mathrm{\Gamma }$ being the quaternion group.