Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

TL;DR

This paper advances the understanding of prime geodesic distribution in 3D hyperbolic spaces, providing improved error bounds and mean subconvexity estimates for specific Kleinian groups and L-functions.

Contribution

It improves the error term bounds in the Prime Geodesic Theorem for the Picard manifold and establishes a mean subconvexity estimate for associated L-functions.

Findings

Improved error bound from O(X^{5/3+ε}) to O(X^{13/8+ε}) for the Picard manifold.

Established a bound of O(X^{16/5+ε}) for the second moment of the error term.

Derived a mean subconvexity estimate for Rankin-Selberg L-functions.

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 $O(X^{16/5+\epsilon})$.