## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### The second moment for counting prime geodesics

Ikuya Kaneko

#### Abstract

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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 96, Number 1 (2020), 7-12.

Dates
First available in Project Euclid: 25 December 2019

https://projecteuclid.org/euclid.pja/1577264417

Digital Object Identifier
doi:10.3792/pjaa.96.002

Mathematical Reviews number (MathSciNet)
MR4047570

#### Citation

Kaneko, Ikuya. The second moment for counting prime geodesics. Proc. Japan Acad. Ser. A Math. Sci. 96 (2020), no. 1, 7--12. doi:10.3792/pjaa.96.002. https://projecteuclid.org/euclid.pja/1577264417

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