Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function
type: Item
instance of: scholarly article
title: Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function
zbMATH Open document ID: 0811.11036
author: Akio Fujii
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-05-02T00:00:00Z
review text: A relation between the average \(\int_ 0^ X \sum_{n=1}^ y \Lambda(n) dy\) of the von-Mangoldt function \(\Lambda(n)\) and the spectrum of the Laplacian for \(L^ 2 (\Gamma \setminus{\mathcal H})\) with \(\Gamma\) the modular group is proved. The proof is based on a Perron formula for the logarithmic derivative of the Selberg zeta-function. The terms involving the zeros of the Riemann zeta-function can then be expressed as a (well-known) Perron integral for the logarithmic derivative of the Riemann zeta-function which produces the von-Mangoldt function.
reviewed by: Roland Matthes
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 559495
zbMATH Keywords: von-Mangoldt function
zbMATH Keywords: spectrum of the Laplacian
zbMATH Keywords: Perron formula
zbMATH Keywords: logarithmic derivative of the Selberg zeta-function
zbMATH Keywords: logarithmic derivative of the Riemann zeta-function
MaRDI profile type: Publication
full work available at URL: pjaa.69.125
OpenAlex ID: W2071227744
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DOI: 10.3792/PJAA.69.125
description: scientific article; zbMATH DE number 559495