A characterization of some unimodular multiplicative functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 0999.11061
publication date: 2001-06-13T00:00:00Z
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 1604212
zbMATH Keywords: characterization
zbMATH Keywords: unimodular multiplicative function
zbMATH Keywords: additive function
author: Bui Minh Phong
MaRDI profile type: Publication
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title: A characterization of some unimodular multiplicative functions
published in: Publicationes Mathematicae Debrecen
review text: An arithmetic function \(g(n)\neq 0\) is said to be a unimodular multiplicative function if \(g(n)\) is multiplicative and \(g(n)\) satisfies the condition \(|g(n)|=1\) for all positive integers \(n\). The author studies unimodular multiplicative functions \(g_1\), \(g_2\) for which \(g_1(an+b)-dg_2(cn)\) tends to zero in some sense. A typical result of this article is the following. NEWLINENEWLINENEWLINETheorem 3: Let \(a\), \(b\), \(c\) be positive integers and let \(d\neq 0\) be a complex number. Then \(g_1\), \(g_2\) unimodular multiplicative functions satisfy the conditions \(g_1(an+b)-dg_2(cn)=o(1)\) where \(n\to \infty\) if and only if there are a real number \(\tau\) and \(G_1\), \(G_2\) unimodular multiplicative functions such that \(g_1(n)=n^{i\tau}G_1(n)\), \(g_2(n)=n^{i\tau}G_2(n)\) and \(G_1(an+b)-d\frac{c^{i\tau}}{a^{i\tau}}G_2(cn)=0\) hold for all positive integers. NEWLINENEWLINENEWLINEFor more information on these latter results, see the paper of \textit{I. Kátai} [Acta. Math. Hung. 58, No. 3/4, 343-350 (1991; Zbl 0746.11036)] and \textit{E. Wirsing, Y. Tang} and \textit{P. Shao} [J. Number Theory 56, 391-395 (1996; Zbl 0845.11030)].
reviewed by: Sándor Turjányi
description: scientific article; zbMATH DE number 1604212