Remarks on the semigroup elements free of large prime factors
The author continues his investigations of elements without large prime factors in arithmetical semigroups, initiated in a recent paper [in: Analytic and probabilistic methods in number theory. Proc. Int. Conf. in Honour of J. Kubilius, Palanga/Lith. 1991, New Trends Probab. Stat. 2, 135-153 (1992; Zbl 0768.11037)], with the aim of developing a theory that is analogous to the classical theory of integers without large prime factors. Let \(G\) be a free commutative semigroup with identity element, generated by a countable subset \(P\) of prime elements, and suppose that there exists a completely additive degree function \(\partial : G \Rightarrow \mathbb{N} \cup \{0\}\) such that \(\partial (p) \geq 1\) for each prime \(p \in P\) and \[ G(n) = \# \bigl \{a \in G : \partial (a) = n \bigr\} = Aq^ n + O(q^{\nu n}) \tag{*} \] for some constants \(A > 0\), \(q > 0\) and \(0 \leq \nu < 1\). Define \[ \Psi (n,m) = \# \bigl\{ a \in G : \partial (a) = n, \quad p | n,\;p \in P \Rightarrow \partial (p) \leq m \bigr\}. \] In the above-quoted paper, the author obtained an estimate for \(\Psi (n,m)\) in terms of the generating function \(Z(y) = \sum^ \infty_{n = 0} G(n)y^ n\) for the range \(m \geq m_ 0\), \(m(\log m) (\log \log m)^ 3 \leq n \leq \exp \{\delta q^ m/m^ 2\}\), where \(m_ 0 \geq 3\) and \(\delta > 0\) are suitable constants. He now shows that the restriction \(n \leq \exp \{\delta q^ m/m^ 2\}\) is unnecessary and may be removed. Moreover, he complements this result by showing that in the range \(m \leq n \ll m^ 2/ \log m\), one has \[ \Psi (n,m) = Aq^ n \rho (u) \left( 1 + \left( {1 + u \log u \over m} \right) \right), \] where \(A\) and \(q\) are as in \((*)\), \(u = n/m\), and \(\rho (u)\) is the Dickman function. This estimate is analogous to the classical Dickman-de Bruijn estimate for the counting function \(\Psi (x,y)\) of integers \(\leq x\) that are free of prime factors exceeding \(y\) and sharpens a recent result of \textit{R. Warlimont} [Manuscr. Math. 71, 197-221 (1991; Zbl 0735.11046)], obtained by a different method.
scientific article; zbMATH DE number 578668