Selberg zeta function

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The Selberg zeta-function was introduced by Atle Selberg in the 1950s. 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.

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 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.)
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 $1/2 - \mathbb{N}$ , and can have zeros or poles at the points $- \mathbb{N}$ .

[edit] Selberg zeta-function for the modular group

For the case where the surface is $\Gamma \backslash \mathbb{H}^2$ , where $Γ$ 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 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 scattering matrix has a pole at $s 0 / 2$ , and hence the Selberg zeta-function has a zero at $s 0 / 2$ .

[edit] Bibliography

Hejhal, D. A. The Selberg trace formula for PSL(2,R). Vol. 2, Springer-Verlag, Berlin, 1983.
Iwaniec, H. Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002.
Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982.