Selberg zeta function

The Selberg zeta-function was introduced by Atle Selberg in the 1950s. It is analogous to the famous Riemann zeta function :$zeta(s)\; =\; prod\_\{pinmathbb\{P\; frac\{1\}\{1-p^\{-s$ 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, $phi(s)$. 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 , 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 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 $s\_0/2$.

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.