Selberg zeta function

The Selberg zeta-function was introduced by. It is analogous to the famous Riemann zeta function
 * $$ \zeta(s) = \prod_{p\in\mathbb{P}} \frac{1}{1-p^{-s}} $$

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,
 * $$\zeta_\Gamma(s)=\prod_p(1-N(p)^{-s})^{-1},$$

or
 * $$Z_\Gamma(s)=\prod_p\prod^\infty_{n=0}(1-N(p)^{-s-n}),$$

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:
 * 1) For every cusp form with eigenvalue $$s_0(1-s_0)$$ there exists a zero at the point $$s_0$$. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the Laplace–Beltrami operator which has Fourier expansion with zero constant term.)
 * 2) The zeta-function also has a zero at every pole of the determinant of the scattering matrix, $$ \phi(s) $$. The order of the zero equals the order of the corresponding pole of the scattering matrix.

The zeta-function also has poles at $$ 1/2 - \mathbb{N} $$, and can have zeros or poles at the points $$ - \mathbb{N} $$.

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 $$ \Gamma \backslash \mathbb{H}^2 $$, 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:
 * $$ \varphi(s) = \pi^{1/2} \frac{ \Gamma(s-1/2) \zeta(2s-1) }{ \Gamma(s) \zeta(2s) }. $$

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 $$s_0/2$$.