On a characterization of some arithmetical multiplicative functions
type: Item
instance of: scholarly article
title: On a characterization of some arithmetical multiplicative functions
zbMATH Open document ID: 0559.10033
DOI: 10.1007/BF01950270
author: Leo Murata
author: Jean-Loup Mauclaire
published in: Acta Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: While a lot is known about regular additive arithmetical functions, the corresponding question for multiplicative functions has been much less investigated. The authors take a step in this direction by showing that if a multiplicative function f satisfies \(\sum_{n\leq x}| f(n+1)- f(n)| =o(g(x))\) but \(\sum_{n\leq x}f(n)\neq o(g(x))\) with a ''nice'' function g, then f is completely multiplicative and \(| f(n)| =n^{\lambda}\) with some constant \(\lambda\geq -1\). To obtain a complete description by showing that \(f(n)=n^ c\) with some complex number c seems to be more difficult and the authors get it only in a very special case. (E. Wirsing has some recent results in this direction.)
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 3891476
zbMATH Keywords: distribution of multiplicative functions
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
cites work: On the distribution function of additive functions
cites work: Q4041656
cites work: On the regularity of arithmetic multiplicative functions. I, II
cites work: On the regularity of arithmetic multiplicative functions. III
description: scientific article; zbMATH DE number 3891476