The second moment for counting prime geodesics

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Ikuya Kaneko¹ 

[1] Tsukuba Kaisei High School

关键词: Prime Geodesic Theorem;    L-functions;    subconvexity;    spectral summation formulæ;    Kloosterman sums;    exponential sums.;   

DOI  :  10.3792/pjaa.96.002

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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.

【 授权许可】

Unknown