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Article

Prime Geodesic Theorem in the Three-dimensional Hyperbolic Space

Transactions of the American Mathematical Society. 2019. Vol. 372. No. 8. P. 5355-5374.
Balkanova O., Chatzakos D., Cherubini G., Frolenkov D., Laaksonen N.

 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 $O(X^{16/5+\epsilon})$.