The second moment for counting prime geodesics. (English) Zbl 1456.11166

Kaneko, Ikuya

Proc. Japan Acad., Ser. A 96, No. 1, 7-12 (2020).

Summary: 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 \(\Gamma = \text{PSL}_2(\mathbb{Z}[i])\), 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 A. Balog et al. [J. Number Theory 198, 239–249 (2019; Zbl 1461.11082)] for \(\Gamma = \text{PSL}_2(\mathbb{Z})\), 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 \(E_{\Gamma}(X) \ll X^{67/42+\varepsilon}\) for \(\Gamma = \text{PSL}_2(\mathbb{Z}[i])\) by an application of the Weyl-type subconvexity.

MSC:

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

Keywords:

prime geodesic theorem; \(L\)-functions; subconvexity; spectral summation formulæ; Kloosterman sums; exponential sums

Citations:

Zbl 1461.11082

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