Let p run over all distinct primitive ordered periodic geodesics, and let τ(p) denote the positive length of p, then every even function h(ρ) analytic in left bracketing bar ℑ[ρ] right bracketing bar <=ϵ + 1/2 and such that left bracketing bar h(ρ) right bracketing bar <=O(( left bracketing bar ρ right bracketing bar )^(-2 - δ)) for ρ-> ± ∞ satisfies the summation formula sum_(k = 0)^∞ h(ρ_k) = (g - 1) integral_(-∞)^∞(-(dh^^)/(d τ))(d τ)/(sinh(1/2 τ)) + sum_({p}) sum_(n = 1)^∞ (τ(p))/(2sinh[1/2 n τ(p)]) h^^(n τ(p)), where g is the genus of the surface whose area is 4π(g - 1) by the Gauss-Bonnet formula.

Selberg's formula | Selberg zeta function