Prime geodesic theorem in the 3-dimensional hyperbolic space

Abstract

For Γ a cofinite Kleinian group acting on H 3 , we study the Prime Geodesic Theorem on M = Γ\H 3 , which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on M . Let E Γ (X) be the error in the counting of prime geodesics with length at most log X. For the Picard manifold, Γ = PSL(2, Z[i]), we improve the classical bound of Sarnak, E Γ (X) = O(X 5/3+ ), to E Γ (X) = O(X 13/8+ ). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg L-function attached to Maass-Hecke cusp forms. We also investigate the second moment of E Γ (X) for a general cofinite group Γ, and show that it is bounded by O(X 16/5+ ).

Second Moment of the Prime Geodesic Theorem for $\mathrm{PSL}(2, \mathbb{Z}[i])$

“…As a corollary of Theorem 1.1 we recover the pointwise bound E Γ (X) ≪ ǫ X 13/8+ǫ of [2, Theorem 1.1]. Furthermore, our second moment bound (1.4) has immediate consequences analogous to Corollary 1.3 and Equation (1.3) in [2], but we will not write them here explicitly. Finally, we observe that Theorem 1.1 implies that the short interval estimate…”

“…By assuming the Lindelöf hypothesis for quadratic Dirichlet L-functions over Gaussian integers, they obtain η = 3/2. It is not clear how far this is from the truth (see the discussion in Remarks 1.5 and 3.1 in [2]).…”

“…Giorgio Schirillo postdoctoral grant (2017)(2018) The estimate (1.2) for the error term is actually valid for all cofinite Kleinian groups, provided that the contribution from possible small eigenvalues is included in the main term. Sarnak's pointwise bound (1.2) has been improved for the Picard group in [15,2,3]. The current best unconditional bound is due to Balkanova and Frolenkov [3], who showed that…”

The second moment of symmetric square L-functions over Gaussian integers

“…and C is the steepest descent contour (see [5, figure 3]). 1 Following [5] we obtain (4.2). It is left to evaluate a 0 , b 0 which are defined by (see [5, (3.31)])…”

“…Since then the function π Γ (X) was intensively studied, see [1,3,9,11]. The currently best known result due to Balog-Biro-Cherubini-Laaksonen [4] states that the error term in (1.1) can be replaced by…”