Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

Olga Balkanova; Dimitrios Chatzakos; Giacomo Cherubini; Dmitry Frolenkov; Niko Laaksonen

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arXiv · 1712.00880

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

Olga Balkanova·Dimitrios Chatzakos·Giacomo Cherubini·Dmitry Frolenkov·Niko Laaksonen

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Abstract

For $Γ$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=Γ\backslash \mathbb{H}^3$, 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, $Γ=\mathrm{PSL}(2,\mathbb{Z}[i])$, 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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Olga Balkanova, Dimitrios Chatzakos, Giacomo Cherubini, Dmitry Frolenkov, Niko Laaksonen. 2018-08-20. Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space. https://arxiv.org/abs/1712.00880

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