Prime geodesic theorem in the 3-dimensional hyperbolic space

Abstract

For Γ \Gamma a cofinite Kleinian group acting on H 3 \mathbb {H}^3 , we study the prime geodesic theorem on M = Γ ∖ H 3 M=\Gamma \backslash \mathbb {H}^3 , which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on M M . Let E Γ ( X ) E_{\Gamma }(X) be the error in the counting of prime geodesics with length at most log ⁡ X \log X . For the Picard manifold, Γ = P S L ( 2 , Z [ i ] ) \Gamma =\mathrm {PSL}(2,\mathbb {Z}[i]) , we improve the classical bound of Sarnak, E Γ ( X ) = O ( X 5 / 3 + ϵ ) E_{\Gamma }(X)=O(X^{5/3+\epsilon }) , to E Γ ( X ) = O ( X 13 / 8 + ϵ ) E_{\Gamma }(X)=O(X^{13/8+\epsilon }) . In the process we obtain a mean subconvexity estimate for the Rankin–Selberg L L -function attached to Maass–Hecke cusp forms. We also investigate the second moment of E Γ ( X ) E_{\Gamma }(X) for a general cofinite group Γ \Gamma , and we show that it is bounded by O ( X 16 / 5 + ϵ ) O(X^{16/5+\epsilon }) .