Arithmetical semigroups. II: Sieving by large and small prime elements. Sets of multiples
type: Item
instance of: scholarly article
title: Arithmetical semigroups. II: Sieving by large and small prime elements. Sets of multiples
zbMATH Open document ID: 0735.11046
DOI: 10.1007/BF02568402
published in: Manuscripta Mathematica
publication date: 1992-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/155613
review text: [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 se…
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 10297
zbMATH Keywords: Dickman function
zbMATH Keywords: Buchstab function
zbMATH Keywords: Mertens formula
zbMATH Keywords: upper and lower densities
zbMATH Keywords: Davenport-Erdős theorem
zbMATH Keywords: Besicovitch theorem
zbMATH Keywords: arithmetical semigroup
zbMATH Keywords: axiom \(A^ \#\)
zbMATH Keywords: linear sieve
zbMATH Keywords: densities of sets of multiples of positive integers
author: Richard Warlimont
reviewed by: John Knopfmacher
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 10297