It is analogous to the famous Riemann zeta function where
The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers.
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
), 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.
is the modular group, the Selberg zeta-function is of special interest.
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