The Selberg zeta-function was introduced by. It is analogous to the famous Riemann zeta function where \mathbb{P} is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If \Gamma is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, or where p runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of \Gamma), and N(p) denotes the length of p (equivalently, the square of the bigger eigenvalue of p). For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface. The zeros are at the following points: The zeta-function also has poles at, and can have zeros or poles at the points. The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function.

Selberg zeta-function for the modular group

For the case where the surface is, where \Gamma is the modular group, the Selberg zeta-function is of special interest. For this special case the Selberg zeta-function is intimately connected to the Riemann zeta-function. In this case the determinant of the scattering matrix is given by: In particular, we see that if the Riemann zeta-function has a zero at s_0, then the determinant of the scattering matrix has a pole at s_0/2, and hence the Selberg zeta-function has a zero at s_0/2.