Open Access
January 2020 The second moment for counting prime geodesics
Ikuya Kaneko
Proc. Japan Acad. Ser. A Math. Sci. 96(1): 7-12 (January 2020). DOI: 10.3792/pjaa.96.002


A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting $E_{\Gamma}(X)$ be the error term arising from counting prime geodesics associated to $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z}[i])$, the bound $E_{\Gamma}(X) \ll X^{3/2+\epsilon}$ is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog \textit{et al.} for $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z})$, and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound $E_{\Gamma}(X) \ll X^{67/42+\epsilon}$ for $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z}[i])$ by an application of the Weyl-type subconvexity.


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Ikuya Kaneko. "The second moment for counting prime geodesics." Proc. Japan Acad. Ser. A Math. Sci. 96 (1) 7 - 12, January 2020.


Published: January 2020
First available in Project Euclid: 25 December 2019

zbMATH: 07192781
MathSciNet: MR4047570
Digital Object Identifier: 10.3792/pjaa.96.002

Primary: 11M36
Secondary: 11F72 , 11L05 , 11M26

Keywords: $L$-functions , exponential sums , Kloosterman sums , prime geodesic theorem , spectral summation formul{æ} , subconvexity

Rights: Copyright © 2020 The Japan Academy

Vol.96 • No. 1 • January 2020
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