The value distribution of differences of additive arithmetic functions
type: Item
instance of: scholarly article
title: The value distribution of differences of additive arithmetic functions
zbMATH Open document ID: 0707.11058
DOI: 10.1016/0022-314X(89)90090-5
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let \(a>0\), \(A>0\), \(b\), \(B\) be integers which satisfy \(aB\ne Ab\) and \(\eta(x)\) a real-valued function defined for \(x\ge 2\). Let \(f_j\), \(j=1,2\), denote real-valued additive functions. In this paper an Erdős-Wintner type criterion is given for the convergence of the distribution functions \[ F_x(z):=[x]^{-1}\#\{1\le n\le x: f_1(an+b)-f_2(An+B)-\eta (x)\le z\}. \] The special case \(a=A=1\), \(b=1\), \(B=0\) and \(\eta(x)\equiv 0\) has been proved in a recent paper by \textit{A. Hildebrand} [Trans. Am. Math. Soc. 310, No. 1, 257--276 (1988; see the preceding review Zbl 0707.11057)]. The arguments used for the proof are mainly based on new results concerning the value distribution of complex multiplicative functions of modulus \(\le 1\).
reviewed by: K.-H. Indlekofer
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4160866
zbMATH Keywords: additive functions
zbMATH Keywords: Erdős-Wintner type criterion
zbMATH Keywords: convergence of the distribution functions
author: Peter D. T. A. Elliott
MaRDI profile type: Publication
cites work: Q3867877
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full work available at URL: 0022-314x(89)90090-5
OpenAlex ID: W2086153299
description: scientific article; zbMATH DE number 4160866