  In this paper we study the analytic torsion for a complete oriented
hyperbolic manifold of finite volume. This requires the definition of a
regularized trace of heat operators. We use the Selberg trace formula to study
the asymptotic behavior of the regularized trace for small time. The main
result of the paper is a new approach to deal with the weighted orbital
integrals on the geometric side of the trace formula.

Spectral Theory