On the mean value of multiplicative functions
type: Item
instance of: scholarly article
title: On the mean value of multiplicative functions
zbMATH Open document ID: 0820.11053
published in: Journal of Number Theory
publication date: 1995-09-14T00:00:00Z
review text: The author generalizes the known mean value theorems for multiplicative functions \(f\) of Delange, Elliot, Daboussi and Indlekofer, but not of Heppner. If there exists a constant \(0\leq c<1\) such that \(\sum_{| f(p)- 1| \leq c} {{f(p)-1} \over p}\) converges, \(\limsup_{s\to 1+} \sum_{| f(p)- 1|\leq c} {{| f(p)| -1} \over p} <\infty\) and there exists a nondecreasing function \(A(x)\) satisfying \(\int_ 1^ \infty {{A(x)} \over {x^ 2}} dx< \infty\), \(| \sum_{n\leq x} h(n)|\leq A(x)\), then \(M(f)\) exists. Here \(h\) is defined by \(f* g^{-1}\) with the completely multiplicative function \[ g(p)= \begin{cases} f(p), \quad &\text{if } | f(p)-1 |\leq c\\ 1, &\text{if } | f(p) -1|>c. \end{cases}. \] This theorem is proved by applying Heppner's theorem on the related function \(g\). The paper ends with two interesting examples, that are covered by this theorem, but not by Heppner's (or even Wirsing's).
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 749188
zbMATH Keywords: arithmetical functions
zbMATH Keywords: mean value theorems
zbMATH Keywords: multiplicative functions
zbMATH Keywords: completely multiplicative function
zbMATH Keywords: Heppner's theorem
author: Wenbin Zhang
reviewed by: Jürgen Spilker
MaRDI profile type: Publication
full work available at URL: jnth.1995.1032
OpenAlex ID: W1980767211
DOI: 10.1006/JNTH.1995.1032
description: scientific article; zbMATH DE number 749188