A characterization of some additive arithmetical functions. V
type: Item
instance of: scholarly article
title: A characterization of some additive arithmetical functions. V
zbMATH Open document ID: 1012.11003
DOI: 10.1023/A:1019708917098
published in: Acta Mathematica Hungarica
publication date: 2003-05-18T00:00:00Z
review text: For Parts I--III see Zbl 0924.11080, Zbl 0969.11033 and Zbl 0969.11034. A real-valued arithmetical function \(f\) is said to be additive if \(f(mn)=f(m)+f(n)\) whenever \((m,n)=1\). \textit{P. Erdős} [Ann. Math. (2) 47, 1-20 (1946; Zbl 0061.07902)] proved that if an additive arithmetical function \(f\) satisfies the condition \(\lim_{n\to\infty}(f(n+1)-f(n))=0\), then \(f(n)=C\log n\) for some constant \(C\). He further conjectured that the conclusion is valid if \(f\) satisfies the condition \(\lim_{x\to\infty}{1\over x}\sum_{n\leq x}|f(n+1)-f(n)|=0\). This was proved by \textit{I. Kátai} [J. Number Theory 2, 1-6 (1970; Zbl 0188.34201)]. These were practically the first of the many articles related to the characterization of the logarithm as an additive arithmetical function. The present article gives a new characterization of the logarithm as an additive arithmetical function.
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 1912069
zbMATH Keywords: logarithm
zbMATH Keywords: additive arithmetical function
zbMATH Keywords: characterization of the logarithm
author: Jean-Loup Mauclaire
MaRDI profile type: Publication
full work available at URL: a:1019708917098
OpenAlex ID: W3015445376
description: scientific article; zbMATH DE number 1912069