The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function
where
or
where p run all over the prime congruent class and N(p) is the norm of congruent class p, which is 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:
- For every cusp form with eigenvalue
s 0 ( 1 − s 0 ) there exists a zero at the points 0 - The zeta-function also has a zero at every pole of the determinant of the scattering matrix,
ϕ ( s ) . The order of the zero equals the order of the corresponding pole of the scattering matrix.
The zeta-function also has poles at
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
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