Mean behaviour and distribution properties of multiplicative functions
type: Item
instance of: scholarly article
title: Mean behaviour and distribution properties of multiplicative functions
zbMATH Open document ID: 1068.11062
author: S. Wehmeier
author: K.-H. Indlekofer
author: Lutz G. Lucht
publication date: 2005-04-07T00:00:00Z
review text: Let \(f: \mathbb{N}\to\mathbb{C}\) be a multiplicative function and \({\mathfrak a}= (a_n)\) a sequence of natural numbers. The function \(f\) has mean-value \(m(f,{\mathfrak a})\) on \({\mathfrak a}\), if the normalized summatory function \(M(f,{\mathfrak a}, x):= {1\over x}\sum_{n\leq x} f(a_n)\) converges to \(M(f,{\mathfrak a})\) as \(x\to\infty\). The authors give criteria for existence of \(M(f,{\mathfrak a})\) and estimates for \(M(f,{\mathfrak a}, x)- M(f,{\mathfrak a})\). They discuss examples as \(a_n= n\), \(a_n= 2^n- 1\) (Mersenne numbers), \(a_n= p_n- 1\) (\(n\)th shifted prime) and visualize their results. Finally, they prove distribution properties and cluster points of the image sequence \(f({\mathfrak a})\).
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 2153535
zbMATH Keywords: Multiplicative functions
zbMATH Keywords: Asymptotic estimates
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Euler products
zbMATH Keywords: Thin sequences
zbMATH Keywords: Uniform summability
zbMATH Keywords: Distribution functions
reviewed by: Jürgen Spilker
MaRDI profile type: Publication
full work available at URL: j.camwa.2004.01.015
OpenAlex ID: W2013434269
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DOI: 10.1016/J.CAMWA.2004.01.015
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 2153535