Strong convergence of additive arithmetical functions
type: Item
instance of: scholarly article
title: Strong convergence of additive arithmetical functions
zbMATH Open document ID: 0578.10054
DOI: 10.1007/BF00966181
published in: Lithuanian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Given an additive function f let \(f_ k\) (k\(\geq 1)\) be the associated ''truncated'' functions defined by \(f_ k(n)=\sum_{p^ m\| n, p\leq k}f(p^ m).\) The author first characterizes those additive functions f, for which the sequence \((f_ k)\) converges strongly to f in the sense that for every \(\epsilon >0\) \[ \lim_{k\to \infty} \limsup_{x\to \infty}(1/x) \#\{n\leq x:\quad | f_ k(n)-f(n)| \geq \epsilon \}=0 ; \] the criterion for this coincides with the Erdős-Wintner criterion for the existence of a limit distribution for f. He then proves similar results involving renormalizing constants. Moreover, he proves an analogue of the law of iterated logarithm for the sequence \((f_ k)\), subject to suitable conditions on the function f.
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3924914
zbMATH Keywords: additive function
zbMATH Keywords: limit distribution
zbMATH Keywords: renormalizing constants
zbMATH Keywords: analogue of the law of iterated logarithm
author: Eugenijus Manstavičius
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
cites work: Probabilistic methods in the theory of numbers
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cites work: Q3867876
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full work available at URL: https://doi.org/10.1007/bf00966181
OpenAlex ID: W2091103452
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description: scientific article; zbMATH DE number 3924914