Proceedings of the Japan Academy, Series A, Mathematical Sciences

The second moment for counting prime geodesics

Ikuya Kaneko

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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

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

First available in Project Euclid: 25 December 2019

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Mathematical Reviews number (MathSciNet)

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula 11L05: Gauss and Kloosterman sums; generalizations 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Prime Geodesic Theorem $L$-functions subconvexity spectral summation formul{æ} Kloosterman sums exponential sums


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.

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