Some number-theoretic identities from group actions.
type: Item
instance of: scholarly article
title: Some number-theoretic identities from group actions.
zbMATH Open document ID: 1187.20015
publication date: 2009-10-26T00:00:00Z
review text: Group actions are applied to some natural situations like the natural `linear' action of \(\text{GL}_r(\mathbb{Z}_n)\) and some of its subgroups to derive number-theoretic identities like \[ \sum_{t_1\in(\mathbb{Z}_n)^*,t_2,\dots,t_r\in\mathbb{Z}_n}\text{GCD}(n,t_1-1,t_2,\dots,t_r)=\varphi(n)\sigma_{r-1}(n). \] Note, that the case \(r=1\) yields the identity \[ \sum_{\text{``\(a\)''}}\text{GCD}(a-1,n)= \varphi(n)d(n), \] where \(d(n)\) stands for the divisor function of \(n\), and where \(\displaystyle\sum_{\text{``\(a\)''}} = \sum_{\text{GCD}(a,n)=1}\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5620617
zbMATH Keywords: not-Burnside lemma
zbMATH Keywords: Cauchy-Frobenius formula
zbMATH Keywords: generalized totient function
zbMATH Keywords: linear actions of finite linear groups
zbMATH Keywords: graph actions
author: Balasubramanian Sury
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: s12215-009-0010-6
OpenAlex ID: W2059655213
cites work: Q4196577
DOI: 10.1007/S12215-009-0010-6
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5620617