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
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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](/images/equations/SelbergTraceFormula/Inline5.svg)
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_({p})sum_(n=1)^infty(tau(p))/(2sinh[1/2ntau(p)])h^^(ntau(p)),](/images/equations/SelbergTraceFormula/NumberedEquation1.svg)
is the genus of the surface whose area is 
by the Gauss-Bonnet
formula.
