High Quality Content by WIKIPEDIA articles! The Selberg zeta-function was introduced by Atle Selberg (1956). 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.

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