Arithmetical semigroups. II: Sieving by large and small prime elements. Sets of multiples
[For part I, cf. ibid. 71, 83-96 (1991; Zbl 0723.11048).] In this paper the author considers an (additive) arithmetical semigroup, i.e. a free monoid G with free generating set \(P\) which admits a ``degree'' map \(\partial:G\to\mathbb{N}\cup\{0\}\) such that \(\partial(1)=0\), \(\partial(p)>0\) for \(p\in P\), \(\partial(ab)=\partial(a)+\partial(b)\), and the number \(G^ \#(n)\) of elements of degree \(n\) is finite for any \(n\geq0\). In part I and most of the rest of the paper, he assumes Axiom \(A^ \#\) of the reviewer's monograph [``Analytic arithmetic of algebraic function fields'' (1979; Zbl 0411.10001)], which requires that \(G^ \#(n)=cq^ n+O(q^{\nu n})\) for constants \(c>0\), \(q>1\), \(\nu<1\). Part I investigates analogues for G of the well known limiting relations of Dickman and Buchstab concerning the number-theoretical functions \[ \psi(x,y)=\#\{n\in\mathbb{N}:n\leq x \hbox{ and } p\mid n\Rightarrow p\leq y\}, \] \[ \phi(x,y)=\#\{n\in\mathbb{N}:n\leq x \hbox{ and } p\mid n\Rightarrow p>y\}, \] which are replaced by analogous functions \(\psi(n,m)=\#\{g\in G:\partial(g)=n \hbox{ and } p\mid g\Rightarrow\partial(p)\leq m\}\), \(\phi(n,m)=\#\{g\in G:\partial(g)=n \hbox{ and } p\mid g\Rightarrow\partial(p)>m\}\). Limiting relations are obtained for these new functions, in terms of the classical Dickman and Buchstab functions \(\rho\) and \(\omega\), respectively, provided in the second case that the generating function \(Z_ G(y)=\sum_{n=0}^ \infty G^ \#(n)y^ n\) has no zero for \(| y|\leq q^{-1}\). In then the pathological case when there is a zero (in which case it must be at \(y=-q^{-1})\), the relation involving \(\omega\) must be replaced by a more complicated one involving two functions \(\omega_ 0\) and \(\omega_ 1\), which happen also to be related to the ``linear sieve'' treated in Chapter 8 of the book by \textit{H. Halberstam} and \textit{H.-E. Richert} [``Sieve methods'' (1974; Zbl 0298.10026)]. Part II of the paper carries over to the above setting some classical results on densities of sets of multiples of positive integers. The densities considered in the present setting are defined as follows: For \(M\subseteq G\), let \(\Delta_ m(n)=\#\{g\in M:\partial(g)=n\}/G^ \#(n)\). Then define \[ \underline{d}(M)=\varliminf\limits_{n\to\infty}\Delta_ M(n),\quad \bar d(M)=\varlimsup\limits_{n\to\infty}\Delta_ M(n),\quad d(M)=\lim_{n\to\infty}\Delta_ M(n) \] if it exists, and \[ \underline\delta(M)=\varliminf\limits_{n\to\infty}{1\over n+1}\sum_{m=0}^ n\Delta_ M(m),\qquad \bar\delta(M)=\varlimsup\limits_{n\to\infty}{1\over n+1}\sum_{m=0}^ n\Delta_ M(m), \] \[ \delta(M)=\lim_{n\to\infty}{1\over n+1}\sum_{m=0}^ n\Delta_ M(m) \] if it exists. Now let \(\phi\neq A\subseteq G\), and let \(M(A)\) denote the set of all multiples in \(G\) of elements of \(A\). Assuming only that \(G^ \#(n)\sim cq^ n\) as \(n\to\infty\), and that \(\sum_{a\in A}q^{-\partial(a)}<\infty\) or the elements of \(A\) are pairwise coprime, the author proves firstly that \(d(M(A))\) exists and equals \(\sup\{d(M(B)):B\subseteq A, B \hbox{ finite}\}\). Two further main results concern analogues of the Davenport- Erdős and Besicovich theorems when \(G\) satisfies Axiom \(A^ \#\): (i) \(\delta(M(A))\) always exists and \(\delta(M(A))=\underline{d}(M(A))=\sup\{d(M(B)):B\subseteq A, B \hbox{ finite}\}\), (ii) for any given \(\epsilon >0\) there exists a set \(A\subseteq G\) with \(\underline{d}(M(A))<\epsilon\) while \(\bar d(M(A))=1\).
densities of sets of multiples of positive integers
scientific article; zbMATH DE number 10297