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 ![|I[rho]|<=epsilon+1/2](http://mathworld.wolfram.com/images/equations/SelbergTraceFormula/Inline5.gif)
and such that 
for 
satisfies the summation formula
![sum_(k=0)^inftyh(rho_k)=(g-1)int_(-infty)^infty(-(dh^^)/(dtau))(dtau)/(sinh(1/2tau))
+sum_(<span style=]() {p})sum_(n=1)^infty(tau(p))/(2sinh[1/2ntau(p)])h^^(ntau(p)), " src="http://mathworld.wolfram.com/images/equations/SelbergTraceFormula/NumberedEquation1.gif" style="height:104px; width:239px" /> |
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.
