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

is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics is a subgroup of SL(2, R), the associated Selberg zeta function is defined as follows,

or

where p runs over conjugacy classes of prime geodesics), and N(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:

1. For every cusp form. 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.)

2. . The order of the zero equals the order of the corresponding pole of the scattering matrix.

.

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

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:

.

See also

• Selberg trace formula

References

• 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.

• 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.