On the density of primitive sets
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 obtained of which we state two: If \(f\) is a smooth weighting and \(\varepsilon>0\), then for all \(N\geq N_0(\varepsilon,f)\) there is a primitive set \(\mathcal A\) such that \(\delta(f,\mathcal A,N)>(1-\varepsilon)/\log\log N\). For every integer \(N\) there is a multiplicative weighting \(f\) satisfying \(f(p)=0\) or 1 and \(f(p^\nu)=0\) for \(\nu\geq2\), such that \(F(f,N)\ll(\log\log N/\log N)^{1/2}\). Moreover, it is conjectured that both of these results are optimal in the sense that in the first case \((1-\varepsilon)\) may not be replaced by \((1+\varepsilon)\) and in the second case \((\log\log N/\log N)^{1/2}\) is the correct order of magnitude of \(F(f,N)\).
scientific article; zbMATH DE number 2136815
scientific article