Spectral estimates for compact hyperbolic space forms and the Selberg zeta function for \(p\)-spectra. II
type: Item
instance of: scholarly article
title: Spectral estimates for compact hyperbolic space forms and the Selberg zeta function for \(p\)-spectra. II
zbMATH Open document ID: 0869.11049
DOI: 10.4171/ZAA/512
author: V. Pereyra
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1997-09-01T00:00:00Z
review text: We maintain the assumptions and the notation of the review of part I of the present work (see Zbl 0868.11047). Let \(S_p\) denote the \(p\)-spectrum of the Laplacian \(\Delta= d\delta +\delta d\) on \(L^2(V)\), and denote by \(d^\delta_p (\mu)\) the dimension of the eigenspace of coclosed \((\delta \alpha= 0)\) eigenforms \(\alpha\) of \(\Delta\) with eigenvalue \(\mu\). Let \(N_p(T)\) denote the sum of the \(d^\delta_p (\mu)\), where the summation extends over all \(\mu\in S_p\) such that \(\mu <T^2+ (p-(n-1)/2)^2\), and let \(c_p: =(4\pi)^{-n/2} {n-1 \choose p} \text{vol} (V)/ \Gamma ({n+2 \over 2})\). Then the main result of the work under review is Theorem B: \(N_p(T)= c_pT^n +O(T^{n-1}/ \log T)\) as \(T\to\infty\). (The reviewer wants to point out that this result is known for \(p=0\) from the work [\textit{J. J. Duistermaat}, \textit{J. A. C. Kolk} and \textit{V. S. Varadarajan}, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52, 27-93 (1979; Zbl 0434.58019), Erratum ibid. 54, 101 (1979; Zbl 0447.58034); see p. 89, Theorem 9.1], where the gamma factor must be corrected to \(\Gamma^{-1} ({n \over 2} +1)\).) The proof of Theorem B uses the author's Selberg trace formula for the \(p\)-spectrum. This formula enables the author to introduce a suitable Selberg zeta function for the \(p\)-spectrum which in its properties closely resembles the well-known case of the Selberg zeta function for a compact Riemann surface. (The case \(n=3\), \(p=0\) was discussed even for cocompact groups with torsion by \textit{J. Elstrodt}, \textit{F. Grunewald} and \textit{J. Mennicke} [The Selberg zeta-function for cocompact discrete subgroups of \(\text{PSL} (2, \mathbb{C})\). Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 83-120 (1985; Zbl 0599.10015)].) The final proof of Theorem B is based on an application of methods of analytic number theory to the Selberg zeta function for the \(p\)-spectrum. The technical details are much …
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 605109
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: length spectrum
zbMATH Keywords: Weyl's law
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/512
OpenAlex ID: W2341008036
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description: scientific article; zbMATH DE number 605109