Selberg zeta function

Abstract

For a geometrically finite hyperbolic surface X the Selberg zeta function ZX(s) was introduced in §2.5. The zeta function is associated with the length spectrum of X (or, equivalently, to traces of conjugacy classes of Γ). We will see in this chapter that it deserves to be thought of as a spectral invariant as well, by virtue of a beautiful correspondence between resonances of X and the zeros of ZX(s).