A note on some characterizations of arithmetic functions
type: Item
instance of: scholarly article
title: A note on some characterizations of arithmetic functions
zbMATH Open document ID: 1186.11005
DOI: 10.1216/RMJ-2009-39-2-645
author: Temba Shonhiwa
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: The author considers the following extensions of the Dirichlet convolution of arithmetical functions: \[ \begin{aligned} G_{(f\circ g)}(n, m) & = \sum_{d|(n,m)}f(d)g(m/d),\\ G_{(h\circ f\circ g)}(n,m) & = \sum_{d|(n,m)} h(d)f(n/d) g(m/d),\end{aligned} \] where each sum is extended over divisors of the gcd of \(n\), \(m\). As might be expected, these sums are multiplicative when each of \(f\), \(g\), \(h\) is multiplicative. A typical characterization result states that a multiplicative \(f\) is completely multiplicative if and only if \[ G_{(f\circ f)}(n,m)= f(m)\sum_{d|(n,m)} 1. \] When \(n= m\) this yields a known theorem of Carlitz.
reviewed by: Tom M. Apostol
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5566193
zbMATH Keywords: convolution
zbMATH Keywords: core function
zbMATH Keywords: multiplicative function
zbMATH Keywords: completely multiplicative function
zbMATH Keywords: quadratic function
MaRDI profile type: Publication
cites work: Some Properties of Completely Multiplicative Arithmetical Functions
cites work: A sorting function
cites work: On an Arithmetic Convolution
cites work: Introduction to arithmetical functions
cites work: On a class of arithmetical convolutions
cites work: Q4040809
cites work: Q5646958
cites work: Q5845454
full work available at URL: rmj-2009-39-2-645
OpenAlex ID: W1987301097
description: scientific article; zbMATH DE number 5566193