Proceedings of the Japan Academy, Series A, Mathematical Sciences

The second moment for counting prime geodesics

Ikuya Kaneko

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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

Permanent link to this document
https://projecteuclid.org/euclid.pja/1577264417

Digital Object Identifier
doi:10.3792/pjaa.96.002

Mathematical Reviews number (MathSciNet)
MR4047570

Subjects
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

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

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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References

  • O. Balkanova, D. Chatzakos, G. Cherubini, D. Frolenkov and N. Laaksonen, Prime geodesic theorem in the 3-dimensional hyperbolic space, Trans. Amer. Math. Soc. 372 (2019), no. 8, 5355–5374.
  • O. Balkanova and D. Frolenkov, Bounds for a spectral exponential sum, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 249–272.
  • O. Balkanova and D. Frolenkov, Sums of Kloosterman sums in the prime geodesic theorem, Q. J. Math. 70 (2019), no. 2, 649–674.
  • A. Balog, A. Biró, G. Cherubini, and N. Laaksonen, Bykovskii-type theorem for the Picard manifold, arXiv:1911.01800.
  • A. Balog, A. Biró, G. Harcos and P. Maga, The prime geodesic theorem in square mean, J. Number Theory 198 (2019), 239–249.
  • V. A. Bykovskiĭ, Density theorems and the mean value of arithmetical functions in short intervals (Russian), Zap. Nauchn. Semin. POMI 212 (1994), 56–70, translation in J. Math. Sci. (N.Y.) 83 (1997), no. 6, 720–730.
  • Y. Cai, Prime geodesic theorem, J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59–72.
  • G. Cherubini and J. Guerreiro, Mean square in the prime geodesic theorem, Algebra Number Theory 12 (2018), no. 3, 571–597.
  • H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159.
  • I. Kaneko, The prime geodesic theorem for $\mathrm{PSL}_{2}(\mathbf{Z}[i])$ and spectral exponential sum, arXiv:1903.05111.
  • I. Kaneko, Spectral exponential sums on hyperbolic surfaces I, arXiv:1905.00681.
  • I. Kaneko and S. Koyama, Euler products of Selberg zeta functions in the critical strip, arXiv:1809.10140.
  • S. Koyama, Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis, Forum Math. 13 (2001), no. 6, 781–793.
  • W. Z. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on $\mathrm{PSL}_2(\mathbf{Z})\backslash \mathbf{H}^{2}$, Inst. Hautes Études Sci. Publ. Math. 81 (1995), no. 1, 207–237.
  • Y. Motohashi, A trace formula for the Picard group. I, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 8, 183–186.
  • Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, 127, Cambridge University Press, Cambridge, 1997.
  • Y. Motohashi, Trace formula over the hyperbolic upper half space, in Analytic number theory (Kyoto, 1996), 265–286, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press, Cambridge, 1997.
  • M. Nakasuji, Prime geodesic theorem via the explicit formula of $\Psi$ for hyperbolic 3-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 130–133.
  • P. Nelson, Eisenstein series and the cubic moment for PGL$_{2}$, arXiv:1911.06310.
  • P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 1, 253–295.
  • A. Selberg, Collected papers, Springer Collected Works in Mathematics, vol. 1, Springer-Verlag, Berlin, Heidelberg, 1989.
  • K. Soundararajan and M. P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120.
  • N. Watt, Spectral large sieve inequalities for Hecke congruence subgroups of $\mathit{SL}(2,\mathbf{Z}[i])$, J. Number Theory 140 (2014), 349–424.
  • H. Wu, Burgess-like subconvexity for, for $\mathrm{GL}_{1}$, Compos. Math. 155 (2019), no. 8, 1457–1499.