Selberg Trace Formula

Let  run over all distinct primitive ordered periodic geodesics, and let  denote the positive length of , then every even function  analytic in  and such that  for  satisfies the summation formula

where  is the genus of the surface whose area is  by the Gauss-Bonnet formula.

REFERENCES:

Balazs, N. L. and Voros, A. "Chaos on the Pseudosphere." Phys. Rep. 143, 109-240, 1986.

Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. http://www.ams.org/notices/200303/fea-conrey-web.pdf.

Elstrodt, J. "Die Selbergsche Spurformel für kompakte Riemannsche Flächen." Jahresber. d. Deutsche Math. Verein 83, 45-77, 1981.

Hejhal, D. A. "The Selberg Trace Formula and the Riemann Zeta Function." Duke Math. J. 43, 441-482, 1976.

Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439-465, 1987.