Selberg zeta function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function where formula_2 is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If formula_3 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 (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: 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.