User:Stratus nebulosus/sandbox/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 primes numbers. If $$\Gamma$$ is a Fuchsian group  (that is, a discrete subgroup of PSL(2,R)) and if $$ \Gamma $$ is finitely generated, the associated Selberg zeta function is defined as
 * $$Z_\Gamma(s)=\prod_p\prod^\infty_{n=0}(1-N(p)^{-s-n}),$$

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). The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function.

Relation to hyperbolic orbifolds
For any geometrically finite  hyperbolic 2-orbifold there is an associated Selberg zeta function. Every hyperbolic 2-orbifold $$ X $$ can be written as a quotient $$ \Gamma\backslash\mathbb{H}^2 $$ where $$ \Gamma $$ is a Fuchsian group (if we assume that $$ \Gamma $$ is torsion-free, then X will be a smooth hyperbolic surface). The set of closed primitive geodesics on $$ X $$ is bijective to set of conjugacy classes of the primitive hyperbolic elements of $$ \Gamma $$. The surface (or orbifold) $$ X $$ is geometric finite if and only if $$ \Gamma $$ is a finitely generated group, in which case the set of primitive closed geodesics is countable. The Selberg zeta function is then defined as
 * $$Z_\Gamma(s)=\prod_p\prod^\infty_{n=0}(1-N(p)^{-s-n}),$$

where p runs over the conjugacy classes of primitive hyperbolic elements of $$\Gamma$$ and $$ N(p) $$ is length of the corresponding closed geodesic. The Selberg zeta function admits a meromorphic continuation to the whole complex plane.

Zeros and Poles
The zeros and poles of the Selberg zeta-function $$ Z_\Gamma(s) $$ can be described in terms of spectral data of the surface $$ X = \Gamma\backslash\mathbb{H}^2. $$

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 $$ - \mathbb{N} $$.

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 $$s_0/2$$.