Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

2017/12/04 by Balkanova, Olga, Chatzakos, Dimitrios, Cherubini, Giacomo +2
#11F72 (Primary) 11M36 #11L05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.00880

Abstract

For Γ a cofinite Kleinian group acting on ℍ³, we study the Prime Geodesic Theorem on M=Γ\backslash ℍ³, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on M. Let E_Γ(X) be the error in the counting of prime geodesics with length at most log X. For the Picard manifold, Γ=PSL(2,ℤ[i]), we improve the classical bound of Sarnak, E_Γ(X)=O(X^5/3+ε), to E_Γ(X)=O(X^13/8+ε). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg L-function attached to Maass-Hecke cusp forms. We also investigate the second moment of E_Γ(X) for a general cofinite group Γ, and show that it is bounded by O(X^16/5+ε).

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