Selberg Zeta Function

The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function
:$\zeta(s) = \prod_ \frac$
where $\mathbb$ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If $\Gamma$ is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows,
:$\zeta_\Gamma(s)=\prod_p(1-N(p)^)^,$
or
:$Z_\Gamma(s)=\prod_p\prod^\infty_(1-N(p)^),$
where ''p'' runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of $\Gamma$), and ''N''(''p'') denotes the length of ''p'' (equivalently, the 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:
# 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 $1/2 - \mathbb$, and can have zeros or poles at the points $- \mathbb$.

The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function.

Selberg zeta-function for the modular group

For the case where the surface is $\Gamma \backslash \mathbb^2$, 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 determinant of the scattering matrix is given by:
:$\varphi(s) = \pi^ \frac.$

In particular, we see that if the Riemann zeta-function has a zero at $s_0$, then the determinant of the scattering matrix has a pole at $s_0/2$, and hence the Selberg zeta-function has a zero at $s_0/2$.

References

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* Iwaniec, H. Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002.
*{{Citation , last1=Selberg , first1=Atle , title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , mr=0088511 , year=1956 , journal=J. Indian Math. Soc. , series=New Series , volume=20 , pages=47–87
* Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982.
* Sunada, T., L-functions in geometry and some applications, Proc. Taniguchi Symp. 1985, "Curvature and Topology of Riemannian Manifolds", Springer Lect. Note in Math. 1201(1986), 266-284.

Zeta and L-functions
Spectral theory