- Selberg zeta function
The Selberg zeta-function was introduced by
Atle Selberg in the 1950s. It is analogous to the famousRiemann zeta function :zeta(s) = prod_{pinmathbb{P frac{1}{1-p^{-s where mathbb{P} is the set of prime numbers. The Selberg zeta-function uses the lengths of simpleclosed geodesic s 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 thecomplex plane . The zeta function is defined in terms of the closedgeodesic s 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 theLaplace-Beltrami operator which hasFourier 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 ackslash 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.
Bibliography
* Hejhal, D. A. The Selberg trace formula for PSL(2,R). Vol. 2, Springer-Verlag, Berlin, 1983.
* 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.
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