Selberg’s zeta functions and surfaces of finite geometry. (Fonctions zêta de Selberg et surfaces de géométrie finie.) (French) Zbl 0794.58044

Kurokawa, N. (ed.) et al., Zeta functions in geometry. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 21, 33-70 (1992).
Summary: Let \(M\) be a Riemann surface of constant curvature \(-1\), finite geometry and totally geodesic compact boundary. With a similar definition to the Selberg’s zeta function associated to a Riemann surface of finite area, the zeta function \(Z_ M\) is expressed through spectral invariants (eigenvalues and resonances) and extends so to a meromorphic function on the entire complex plane. Linked to trace formulas of Selberg’s and Birman-Krejn’s type, the proof is based on the meromorphic extensions of the resolvent of various Laplacians and the following (stationary and non-stationary) scattering theory.
For the entire collection see [Zbl 0771.00036].


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11F72 Spectral theory; trace formulas (e.g., that of Selberg)