The Selberg trace formula and the Selberg zeta-function for cocompact discrete subgroups of \(SO_ +(1,n)\) (Q1324674)

Selberg's trace formula is derived for \(n\)-dimensional hyperbolic space in the cocompact case. The author follows the classical setup. Most parts of the calculations are omitted. Choosing a suitable test-function in the trace formula, the resulting loxodromic term is shown to be a logarithmic derivative of a meromorphic function which is then defined to be Selberg's zeta-function. For low dimensions this coincides with the usual definition. Analytic properties such as a functional equation are derived from the trace formula; a prime geodesic theorem follows in a standard way.