Prime geodesic theorem via the explicit formula of \(\Psi\) for hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Prime geodesic theorem via the explicit formula of \(\Psi\) for hyperbolic 3-manifolds
zbMATH Open document ID: 1028.11032
author: Maki Nakasuji
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-09-12T00:00:00Z
review text: Let \(\Gamma< \text{PSL} (2,\mathbb{C})\) be a cofinite group acting on the upper half-space model \(\mathbb{H}\) of hyperbolic three-space, and let \(\pi_\Gamma(x)\) denote the number of associated prime geodesics of length at most \(\log x\), i.e., the number of primitive hyperbolic or loxodromic conjugacy classes \(\{P_0\}\) in \(\Gamma\) satisfying \(N(P_0)\leq x\) (where \(N(P_0)\) denotes the norm of \(P_0\)). Denoting by \(s_0=2\), \(s_1,\dots, s_M\) the \(s\)-values belonging to the small eigenvalues \(0= \lambda_0, \lambda_1,\dots, \lambda_M\leq 1\) of the Laplacian on \(\Gamma \setminus \mathbb{H}\) the function \(\pi_\Gamma\) satisfies an asymptotic law of the form \[ \pi_\Gamma(x)= \text{li}(x^2)+ \sum_{k=1}^M \text{li}(x^{s_k})+ R(x) \] where li denotes the logarithmic integral and where \(R(x)\) denotes an error term. When \(\Gamma\) is cocompact or when \(\Gamma\) is cofinite and satisfies an extra condition the author shows that \[ R(x)= \Omega_\pm \Biggl( \frac {x(\log\log x)^{1/3}} {\log x}\Biggr) \quad\text{as}\quad x\to \infty. \] The proof is based on the Selberg zeta-function and the usual techniques applied in the proof of the prime number theorem. In this way the author also proves: When \(\Gamma\) is cocompact or \(\Gamma= \text{PSL} (2,{\mathfrak o}_K)\) with \(K\) an imaginary quadratic number field, then \[ R(x)= O(x^{(5/3)+ \varepsilon}) \quad\text{as}\quad x\to \infty. \]
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1800027
zbMATH Keywords: prime geodesic theorem
zbMATH Keywords: hyperbolic three-manifold
zbMATH Keywords: Selberg zeta-function
MaRDI profile type: Publication
full work available at URL: pjaa.77.130
OpenAlex ID: W2091236617
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DOI: 10.3792/PJAA.77.130
description: scientific article; zbMATH DE number 1800027