Selberg zeta function


The Selberg zeta-function was introduced by. It is analogous to the famous Riemann zeta function
where is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If is a subgroup of SL, the associated Selberg zeta function is defined as follows,
or
where p runs over conjugacy classes of prime geodesics, and N denotes the length 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, can be described in terms of spectral data of the surface.
The zeros are at the following points:
  1. For every cusp form with eigenvalue there exists a zero at the point. The order of the zero equals the dimension of the corresponding eigenspace.
  2. The zeta-function also has a zero at every pole of the determinant of the scattering matrix,. The order of the zero equals the order of the corresponding pole of the scattering matrix.
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 analogue of the Selberg zeta function.

Selberg zeta-function for the modular group

For the case where the surface is, where 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, then the determinant of the scattering matrix has a pole at, and hence the Selberg zeta-function has a zero at.