A New Interpretation of the Selberg Trace Formula

P. Cartier and A. Voros


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.

Selberg trace formula

Math Review Classification
Primary 11M36

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8 pages

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