Analytic extensions of the zeta functions for surfaces of variable negative curvature (Q1121576)

The main aim of the present paper is to prove that the zeta-function of a compact \(C^{\infty}\)-Riemannian surface S of variable negative curvature is non-zero and holomorphic in a half-plane \(Re(s)>h-\delta\) \((\delta >0)\) except for a simple pole at \(s=h\). Here, \(h>0\) denotes the topological entropy of the geodesic flow on the unit tangent bundle of S. The proof is based on a study of this geodesic flow. Corollary 1: The Selberg zeta-function for a compact \(C^{\infty}\)-Riemannian surface of variable negative curvature is non-zero and holomorphic for \(Re(s)>h- \delta\) \((\delta >0)\) except for a simple zero at \(s=h\). - The consequences of these results for the asymptotic behaviour of the geodesic flow will be discussed in a future publication by the author.