Holomorphic extensions of trace formulas

The Chazarain-Poisson summation formula for Riemannian manifolds (which generalizes the Poisson Summation formula)
computes the distribution trace. In the case of Riemannian surfaces with constant (sectional) curvature, we study
the holomorphic extension of the shifted trace. We have three generic cases according to the sign of the curvature:
the sphere, the torus and the compact hyperbolic surfaces of negative constant curvature. We use the shifted
Laplacian in order to be able to use the Selberg trace formula. Our results concern the case of the torus, the case
of a compact Riemannian surfaces with constant (sectional) negative curvature, and the case of a compact Riemannian
manifold of dimension $n$ and constant curvature, $n\ge 3$.