The sum of \(\alpha^{\Omega(n)}\) over integers n\(\leq x\) with all prime factors between \(\alpha\) and y
type: Item
instance of: scholarly article
title: The sum of \(\alpha^{\Omega(n)}\) over integers n\(\leq x\) with all prime factors between \(\alpha\) and y
zbMATH Open document ID: 0531.10045
DOI: 10.1016/0022-314X(84)90054-4
author: Douglas Hensley
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: Estimates of the function \(\psi_{\alpha}(x,y)=\sum \alpha^{\Omega(n)}\), \(\alpha\geq 1\), where the sum is taken over all n, 1\(\leq n\leq x\) such that if p is prime and \(p| n\) then \(\alpha<p\leq y\), are discussed. This function is related to the well known function \(\psi\) (x,y), which represents the number of \(n\leq x\) all of whose prime factors are \(\leq y\). The function by which \(\psi_{\alpha}(x,y)\) is approximated is \(x\rho_{\alpha}(u)\log^{\alpha -1}x\), where \(u=\log x/\log y\) and \(\rho_{\alpha}(u)\) is a solution of a certain differential delay equation. \(\{\) Reviewer's remark: in the definition of \(\rho_{\alpha}\) on p. 207 in ii) one should replace \(<\) by \(=\}\).
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3841984
zbMATH Keywords: Dickman-de Bruijn function
zbMATH Keywords: largest prime factor
zbMATH Keywords: asymptotic behaviour
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3841984