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The prime geodesic theorem for $\operatorname{PSL}_{2}(\mathbb{Z}[i])$ and spectral exponential sums

Ikuya Kaneko

Vol. 16 (2022), No. 8, 1845–1887
Abstract

This work addresses the prime geodesic theorem for the Picard manifold = PSL 2([i]) 𝔥3, which asks for the asymptotic evaluation of a counting function for the closed geodesics on . Let EΓ(X) be the error term in the prime geodesic theorem. We establish that EΓ(X) = O𝜖(X32+𝜖) on average as well as many pointwise bounds. The second moment bound parallels an analogous result for Γ = PSL 2() due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet L-functions over (i). Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group Γ is given. Our numerical experiments visualise in particular that EΓ(X) obeys a conjectural bound of size O𝜖(X1+𝜖).

Keywords
prime geodesic theorem, Picard manifold, second moment, $L$-functions, Selberg trace formula, Kuznetsov formula, Kloosterman sums, spectral exponential sums, subconvexity
Mathematical Subject Classification
Primary: 11M36
Secondary: 11F72, 11L05, 11M26
Milestones
Received: 28 March 2020
Revised: 17 February 2021
Accepted: 1 April 2021
Published: 29 November 2022
Authors
Ikuya Kaneko
The Division of Physics, Mathematics and Astronomy
California Institute of Technology
Pasadena, CA
United States