Integral Representations of the Logarithmic Derivative of the Selberg Zeta Function

Abstract

We point out the importance of the integral representations of the logarithmic derivative of the Selberg zeta function valid up to the critical line, i.e. in the region that includes the right half of the critical strip, where the Euler product definition of the Selberg zeta function does not hold. Most recent applications to the behavior of the Selberg zeta functions associated to a degenerating sequence of finite volume, hyperbolic manifolds of dimension 2 and 3 are surveyed. The research problem consists in extending this kind of integral representations to the setting of the locally symmetric spaces of rank 1.