On the density of primitive sets
type: Item
instance of: scholarly article
title: On the density of primitive sets
zbMATH Open document ID: 1068.11060
author: Levon H. Khachatrian
author: András Sárközy
author: Rudolf Ahlswede
published in: Journal of Number Theory
publication date: 2005-02-22T00:00:00Z
review text: A set of positive integers \(\mathcal A\) is said to be primitive if no element of \(\mathcal A\) divides any other element of \(\mathcal A\). Investigations of primitive sets center around questions about their densities with respect to various weighting functions \(f\). More precisely, given a nonnegative arithmetic function \(f\) the \(f\)-density of \(\mathcal A\) at \(N\) is defined by \(\delta(f,\mathcal A,N)=\sum_{a\leq N,\;a\in\mathcal A} f(a) \big/\sum_{n\leq N}f(n)\). Thus, for example, the upper asymptotic density of \(\mathcal A\) is given by \(\limsup_{N\to\infty}\delta(f,\mathcal A,N)\), with \(f\) identically 1, while the lower logarithmic density of \(\mathcal A\) is given by \(\liminf_{N\to\infty}\delta(f,\mathcal A,N)\), with \(f(n)=1/n\). These two choices of the weighting function \(f\) together with the third choice, \(f(n)=1/(n\log n)\), have been the subject of a number of investigations [see, for example, \textit{H. Halberstam} and \textit{K. F. Roth}, Sequences. Berlin: Springer (1983; Zbl 0498.10001)]. In this paper the authors investigate \(\delta(f,\mathcal A,N)\), and especially \(F(f,N)=\max_{\mathcal A}\delta(f,\mathcal A,N)\), for several families of weighting functions \(f\). One such family is given by \(f_\sigma(n)=n^{-\sigma}\), with \(0\leq\sigma<\infty\). In this case the order of magnitude of \(F(f_\sigma,N)\) is determined as follows: \(F(f_\sigma,N)\asymp1\), for \(0\leq\sigma\leq9/10\) and \(\sigma\geq11/10\); \(F(f_\sigma,N)\asymp(\log\log N)^{-1/2}\), for \(| \sigma-1| \leq3/\log N\); and \(F(f_\sigma,N)\asymp(\log1/| \sigma-1| )^{-1/2}\), for \(\sigma\) in the remaining range. More generally, the authors consider arbitrary nonnegative multiplicative functions \(f\), as well as \textit{smooth} weights \(f\) which are defined by the conditions: \(0\leq f(n)\leq1\) and, for some \(n_0\), we have \(f(n_0)>0\) and \(f(n)\leq f(n-1)\) for \(n>n_0\). For these general families of weighting functions \(f\) a number of results are…
reviewed by: Gennady Bachman
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 2136815
zbMATH Keywords: primitive sets
zbMATH Keywords: density
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.07.011
OpenAlex ID: W2068465902
cites work: Q2785576
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DOI: 10.1016/J.JNT.2004.07.011
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description: scientific article; zbMATH DE number 2136815