The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function 
formula_1 
where formula_2 is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If formula_3 is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows,
formula_4
or
formula_5
where "p" runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of formula_3), 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:
The zeta-function also has poles at formula_10, and can have zeros or poles at the points formula_11.
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 formula_12, where formula_13 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:
formula_14
In particular, we see that if the Riemann zeta-function has a zero at formula_8, then the determinant of the scattering matrix has a pole at formula_16, and hence the Selberg zeta-function has a zero at formula_16.