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.
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. 07121875 10.1090/tran/7720O. 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. 07121875 10.1090/tran/7720
O. Balkanova and D. Frolenkov, Bounds for a spectral exponential sum, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 249–272. MR3939255 07053661 10.1112/jlms.12174O. Balkanova and D. Frolenkov, Bounds for a spectral exponential sum, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 249–272. MR3939255 07053661 10.1112/jlms.12174
O. Balkanova and D. Frolenkov, Sums of Kloosterman sums in the prime geodesic theorem, Q. J. Math. 70 (2019), no. 2, 649–674. MR3975527 07086680 10.1093/qmath/hay060O. Balkanova and D. Frolenkov, Sums of Kloosterman sums in the prime geodesic theorem, Q. J. Math. 70 (2019), no. 2, 649–674. MR3975527 07086680 10.1093/qmath/hay060
A. Balog, A. Biró, G. Cherubini, and N. Laaksonen, Bykovskii-type theorem for the Picard manifold, arXiv:1911.01800. 1911.01800A. Balog, A. Biró, G. Cherubini, and N. Laaksonen, Bykovskii-type theorem for the Picard manifold, arXiv:1911.01800. 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. 07005769 10.1016/j.jnt.2018.10.012A. Balog, A. Biró, G. Harcos and P. Maga, The prime geodesic theorem in square mean, J. Number Theory 198 (2019), 239–249. 07005769 10.1016/j.jnt.2018.10.012
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.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.
G. Cherubini and J. Guerreiro, Mean square in the prime geodesic theorem, Algebra Number Theory 12 (2018), no. 3, 571–597. MR3815307 1422.11122 10.2140/ant.2018.12.571 euclid.ant/1532743365G. Cherubini and J. Guerreiro, Mean square in the prime geodesic theorem, Algebra Number Theory 12 (2018), no. 3, 571–597. MR3815307 1422.11122 10.2140/ant.2018.12.571 euclid.ant/1532743365
H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159. MR743969 0527.10021H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159. MR743969 0527.10021
I. Kaneko, The prime geodesic theorem for $\mathrm{PSL}_{2}(\mathbf{Z}[i])$ and spectral exponential sum, arXiv:1903.05111. 1903.05111I. Kaneko, The prime geodesic theorem for $\mathrm{PSL}_{2}(\mathbf{Z}[i])$ and spectral exponential sum, arXiv:1903.05111. 1903.05111
I. Kaneko, Spectral exponential sums on hyperbolic surfaces I, arXiv:1905.00681. 1905.00681I. Kaneko, Spectral exponential sums on hyperbolic surfaces I, arXiv:1905.00681. 1905.00681
I. Kaneko and S. Koyama, Euler products of Selberg zeta functions in the critical strip, arXiv:1809.10140. 1809.10140I. Kaneko and S. Koyama, Euler products of Selberg zeta functions in the critical strip, arXiv:1809.10140. 1809.10140
S. Koyama, Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis, Forum Math. 13 (2001), no. 6, 781–793. MR1861249 1061.11024 10.1515/form.2001.034S. Koyama, Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis, Forum Math. 13 (2001), no. 6, 781–793. MR1861249 1061.11024 10.1515/form.2001.034
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. MR1361757 10.1007/BF02699377W. 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. MR1361757 10.1007/BF02699377
Y. Motohashi, A trace formula for the Picard group. I, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 8, 183–186. 0876.11021 10.3792/pjaa.72.183 euclid.pja/1195510241Y. Motohashi, A trace formula for the Picard group. I, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 8, 183–186. 0876.11021 10.3792/pjaa.72.183 euclid.pja/1195510241
Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, 127, Cambridge University Press, Cambridge, 1997. 0878.11001Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, 127, Cambridge University Press, Cambridge, 1997. 0878.11001
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. 0910.11021Y. 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. 0910.11021
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. 1028.11032 10.3792/pjaa.77.130 euclid.pja/1148393038M. 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. 1028.11032 10.3792/pjaa.77.130 euclid.pja/1148393038
P. Nelson, Eisenstein series and the cubic moment for PGL$_{2}$, arXiv:1911.06310. 1911.06310P. Nelson, Eisenstein series and the cubic moment for PGL$_{2}$, arXiv:1911.06310. 1911.06310
P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 1, 253–295. 0527.10022 10.1007/BF02393209 euclid.acta/1485890267P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 1, 253–295. 0527.10022 10.1007/BF02393209 euclid.acta/1485890267
A. Selberg, Collected papers, Springer Collected Works in Mathematics, vol. 1, Springer-Verlag, Berlin, Heidelberg, 1989. 0675.10001A. Selberg, Collected papers, Springer Collected Works in Mathematics, vol. 1, Springer-Verlag, Berlin, Heidelberg, 1989. 0675.10001
K. Soundararajan and M. P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120. MR3028757 1276.11084K. Soundararajan and M. P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120. MR3028757 1276.11084
N. Watt, Spectral large sieve inequalities for Hecke congruence subgroups of $\mathit{SL}(2,\mathbf{Z}[i])$, J. Number Theory 140 (2014), 349–424. 1312.11032 10.1016/j.jnt.2014.01.018N. Watt, Spectral large sieve inequalities for Hecke congruence subgroups of $\mathit{SL}(2,\mathbf{Z}[i])$, J. Number Theory 140 (2014), 349–424. 1312.11032 10.1016/j.jnt.2014.01.018
H. Wu, Burgess-like subconvexity for, for $\mathrm{GL}_{1}$, Compos. Math. 155 (2019), no. 8, 1457–1499. MR3977317 07077744 10.1112/S0010437X19007309H. Wu, Burgess-like subconvexity for, for $\mathrm{GL}_{1}$, Compos. Math. 155 (2019), no. 8, 1457–1499. MR3977317 07077744 10.1112/S0010437X19007309
