On the distribution in short intervals of integers having no large prime factor
type: Item
instance of: scholarly article
title: On the distribution in short intervals of integers having no large prime factor
zbMATH Open document ID: 0606.10033
DOI: 10.1016/0022-314X(87)90031-X
author: J. B. Friedlander
author: Jeffrey C. Lagarias
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Psi\) (x,y) be the number of integers \(\leq x\) all of whose prime factors are \(\leq y\). The authors investigate the function \(\Psi (x, x^{\alpha})-\Psi (x-x^{\beta}, x^{\alpha})\) with particular regard to when it is greater than zero for large x and when it is greater than \(c(\alpha,\beta)x^{\beta}\) for some positive constant c(\(\alpha\),\(\beta)\) for large x. Define f(\(\alpha)\) to be the infimum of the values of \(\beta\) for which, for all \(\alpha_ 1>\alpha\), it is true that \(\Psi (x, x^{\alpha_ 1})-\Psi (x-x^{\beta}, x^{\alpha_ 1})>0\) for large x and define \(f^*(\alpha)\) similarly where this difference is greater than \(c(\alpha_ 1,\beta)x^{\beta}\). The authors show that there exists a positive constant c such that \(f^*(\alpha)\leq 1-\alpha -c\alpha (1-\alpha)^ 3\) for \(0\leq \alpha \leq 1\), and that \(f(\alpha)\leq 1-2\alpha (1-2^{-[\alpha^{-1}]})\) for \(0<\alpha \leq 1/2\). They also show that for fixed \(\alpha\) and \(\beta\) and almost all x, \[ \Psi (x, x^{\alpha})-\Psi (x-x^{\beta}, x^{\alpha})\geq (1/64)\quad \beta \rho (1/\alpha) x^{\beta}, \] where \(\rho\) (t) is Dickman's function.
reviewed by: W. E. Briggs
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3979171
zbMATH Keywords: integers without large prime factors
zbMATH Keywords: short intervals
zbMATH Keywords: Dickman's function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3979171