Selberg zeta function

The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function

where ${\displaystyle \mathbb {P} }$ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers.　If ${\displaystyle \Gamma }$ is a subgroup of SL(2,R) Selberg zeta function is defined as follows,

or

where p run all over the prime congruent class and N(p) is the norm of congruent class p, which is 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:

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

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