Meromorphic Continuation of the Spectral Zeta Kernel

The meromorphic extension of spectral zeta kernel, (g(s,x,x), of the Laplacian on a Riemannian manifold, (M,g), is constructed using a relationship between the zeta kernel and the heat kernel. This construction is facilitated by the analytic properties of the Mellin transform of measurable functions on Riemannian manifolds. The pole structure of the spectral zeta function is highlighted from the relationship between it and the zeta kernel.