The Thermodynamic Formalism Approach to Selberg's Zeta Function for Psl(2, Z)

An Extension of the Thermodynamic Formalism Approach to Selberg's Zeta Function for General Modular Groups

Selberg zeta functions for spaces of higher rank

5 Introduction In 1956 A. Selberg introduced the zeta function Z(s) = c N ≥0 (1 − e −(s+N)l(c)), Re(s) >> 0, where the first product is taken over all primitive closed geodesics in a compact Riemannian surface of genus ≥ 2, equipped with the hyperbolic metric, and l(c) denotes the length of the geodesic c. Selberg proved that the product converges if the real part of s is large enough and that ...

On the Zeros of Maass Wave Form L-functions

In [1], Epstein, Sarnak, and I gave some general results concerning the zeros of an L-function attached to an even Maass wave form for T = PSL(2, Z). The main result was that such an L-function has many zeros on its critical line. In fact, we showed that the analogue of the Hardy-Littlewood theorem for the Riemann zeta function holds for these L-functions as well. In this note I announce the ne...

The Selberg Trace Formula and Selberg Zeta-Function for Cofinite Kleinian Groups with Finite Dimensional Unitary Representations

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...

The Divisor of Selberg's Zeta Function for Kleinian Groups

We compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups acting on a real hyperbolic space of dimension n + 1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = nH together with the Euler characteristic of X compacti ed to a manifold with boundary. If n is even, the singularities of the zeta funciton associated...