An elementary proof of the Petersson-Knopp theorem on Dedekind sums
type: Item
instance of: scholarly article
title: An elementary proof of the Petersson-Knopp theorem on Dedekind sums
zbMATH Open document ID: 0444.10006
DOI: 10.1016/0022-314X(80)90044-X
author: Larry A. Goldberg
published in: Journal of Number Theory
publication date: 1980-01-01T00:00:00Z
review text: \textit{M. I. Knopp} [J. Number Theory 12, 2--9 (1980; Zbl 0423.10015)] used the functional equation of Dedekind's eta function together with Hecke operators to show that the classical Dedekind sum \(s(h,k)\) satisfies the identity \[ \sum_{_{\substack{ ad=n \\ d>0}}} \sum_{b\pmod d} s(ah+bk,dk)=\sigma(n) s(h,k) \] for arbitrary integers \(n>0\), \(h\) and \(k>0\), where \(\sigma(n)\) is the sum of the divisors of \(n\). The author gives a short elementary proof based on an identity of \textit{P. Subrahmanyam} [Math. Stud. 45, 8--12 (1977)] which expresses the inner sum as follows: \[ \sum_{b\pmod d} s(ah + bk, dk) = \sum_{c\mid d} \mu(c) s(ahc,k)\sigma(d/c). \] When this is summed over all \(d>0\) with \(ad = n\) the right member collapses to \(\sigma(n) s(h,k)\). A completely different elementary proof has been given by \textit{L. A. Parson} [Math. Proc. Camb. Philos. Soc. 88, 11--14 (1980; Zbl 0435.10005)].
Mathematics Subject Classification ID: 11F20
zbMATH DE Number: 3693353
zbMATH Keywords: Petersson-Knopp theorem
zbMATH Keywords: Dedekind sums
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
cites work: Hecke operators and an identity for the Dedekind sums
cites work: Dedekind sums and Hecke operators
cites work: Q3947715
description: scientific article; zbMATH DE number 3693353