The prime geodesic theorem for \(\mathrm{PSL}2(\mathbb{Z}[i])\) and spectral exponential sums
type: Item
instance of: scholarly article
title: The prime geodesic theorem for \(\mathrm{PSL}2(\mathbb{Z}[i])\) and spectral exponential sums
author: Ikuya Kaneko
published in: Algebra \& Number Theory
publication date: 2022-12-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1903.05111
review text: This article deals with the prime geodesic theorem on the Picard manifold \(\Gamma\backslash\mathfrak h^3\) with \(\mathfrak h^3\) the 3-dimensional upper half space and \(\Gamma=\mathrm{PSL}(2,\mathbb Z[\sqrt{-1}])\). Let \[ \Psi_\Gamma(X)=\sum_{N(P)\le X}\log N(P_0), \] where \(P\) runs through hyperbolic and loxodromic conjugacy classes of \(\Gamma\), \(P\) is a power of a primitive class \(P_0\), and \(N(P)\) stands for its norm. The main purpose of this article is to improve the error term \[ E_\Gamma(X)=\Psi_\Gamma(X)-\sum_{1<s_j\le2}\frac{X^{is_j}}{s_j} \] of prime geodesic theorem with \(s_j\) the exceptional zeros of the Selberg zeta function of \(\Gamma\). It is known that subconvexity estimates of automorphic \(L\)-functions play an essential role for improving \(E_\Gamma(X)\). Indeed \textit{S.-y. Koyama} [Forum Math. 13, No. 6, 781--793 (2001; Zbl 1061.11024)] proved that \(E_\Gamma(X)=O_\varepsilon(X^{11/7+\varepsilon})\) under the mean-Lindelöf hypothesis for symmetric square \(L\)-functions associated to the Hecke-Maass forms, which was a conditional improvement of the trivial bound \(O(X^{5/3+\varepsilon})\). In the article under review the author succeeds in obtaining an unconditional bound \[ E_\Gamma(X)=O_\varepsilon(X^{376/237+\varepsilon}) \] as well as in clarifying the relation between the subconvexity estimates and the improvement of \(E_\Gamma(X)\). He also presents a better estimate of \(S(T, X)=\sum_{t_j\le T}X^{it_j}\) with \(s_j=1+it_j\) in the spectral aspect.
reviewed by: Shin-ya Koyama
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 7628581
zbMATH Keywords: prime geodesic theorem
zbMATH Keywords: Picard manifold
zbMATH Keywords: second moment
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: Kuznetsof formula
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: spectral exponential sums
zbMATH Keywords: subconvexity
MaRDI profile type: Publication
OpenAlex ID: W4311032041
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DOI: 10.2140/ANT.2022.16.1845
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