The algebraic independence of the sum of divisors functions
type: Item
instance of: scholarly article
title: The algebraic independence of the sum of divisors functions
zbMATH Open document ID: 1214.11011
author: Daniel Lustig
published in: Journal of Number Theory
publication date: 2010-10-19T00:00:00Z
review text: Let \(\sigma_j(n)=\sum_{d|n}d^j\) be the sum of divisors function, and let \(I\) be the identity function. The author shows that the set of functions \(\{\sigma_i\}_{i=0}^\infty\cup \{I\}\) is algebraically independent. Let \(\pi\) be a prime, \(s\) be an integer relatively prime to \(\pi\), \(\alpha\) be a nonnegative integer, \(r=\pi^\alpha s\) and \(n=\pi^{2\alpha+1}s\), and \(j\) be a fixed, nonnegative integer. The author proves that the identity \[ n^j\sigma_j(r)+r^j\sigma_j(r)=r^j\sigma_j(n) \] is the unique algebraic identity in the variables \(n,r,\sigma_j(r)\), and \(\sigma_j(n)\), up to multiplication by a constant. Furthermore, he proves that any perfect number \(n\) must have the form \(n=\frac{r\sigma(r)}{2r-\sigma(r)}\), with some restrictions on \(r\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5802573
zbMATH Keywords: sum of divisors
zbMATH Keywords: algebraic independence
zbMATH Keywords: perfect numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.03.022
OpenAlex ID: W2041620078
cites work: The algebraic independence of arithmetic functions. I: Multiplicative functions
cites work: Improved Techniques for Lower Bounds for Odd Perfect Numbers
cites work: Conditions Equivalent to the Existence of Odd Perfect Numbers
DOI: 10.1016/J.JNT.2010.03.022
description: scientific article; zbMATH DE number 5802573