We extend the Selberg trace formula for a hyperbolic compact Riemann surface to some new test functions, i.e., holomorphic and decreasing at infinity in a sector instead of a horizontal strip (and no longer even). As applications: 1) we interpret the trace formula as a Poisson summation formula involving the eigenvalue spectrum of the hyperbolic Laplacian on one side, and the lengths of all (real and complex) periodic geodesics of the surface on the other side; 2) we obtain a closed meromorphic continuation formula for a spectral zeta function of the hyperbolic Laplacian. |

__Keywords__

Selberg trace formula

__Math Review Classification__

Primary 11M36

__Last Updated__

__Length__

8 pages

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