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

摘要

In this paper, we establish the formulas expressing the special values of the spectral zeta function ζ_Δ(n) of the Laplacian Δ on some locally symmetric Riemannian manifold ΓG/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 Δ by computing the values ζ_Δ(n) numerically, when ΓG/K is a Riemann surface with Γ being the quaternion group.