Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

Abstract

For $\Gamma$ a cofinite Kleinian group acting on ${\mathbb{H}}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash {\mathbb{H}}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\Gamma }(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $\Gamma =\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_{\Gamma }(X)=O(X^{5/3+\epsilon })$, to $E_{\Gamma }(X)=O(X^{13/8+\epsilon })$. 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 $E_{\Gamma }(X)$ for a general cofinite group $\Gamma$, and show that it is bounded by $O(X^{16/5+\epsilon })$.

Corrected proof of Theorem 3.3 (with a weaker bound), added two authors, 18 pages