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The Selberg zeta-function was introduced by Atle Selberg (1956). 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, \( \phi(s) \) . 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 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 \( s_0/2 \).
References

Fischer, Jürgen (1987), An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Mathematics, 1253, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0077696, ISBN 978-3-540-15208-8, MR892317
Hejhal, Dennis A. (1976), The Selberg trace formula for PSL(2,R). Vol. I, Lecture Notes in Mathematics, Vol. 548, 548, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079608, MR0439755
Hejhal, Dennis A. (1983), The Selberg trace formula for PSL(2,R). Vol. 2, Lecture Notes in Mathematics, 1001, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0061302, ISBN 978-3-540-12323-1, MR711197
Iwaniec, H. Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002.
Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc. (N.S.) 20: 47–87, MR0088511
Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982.

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