Sur le nombre des entiers sans grand facteur premier. (On the number of integers without large prime factor)
type: Item
instance of: scholarly article
title: Sur le nombre des entiers sans grand facteur premier. (On the number of integers without large prime factor)
zbMATH Open document ID: 0676.10028
DOI: 10.1016/0022-314X(89)90099-1
author: Éric Saïas
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: An estimate for \(\psi(x,y)\), the number of positive integers not exceeding \(x\) with no prime factor exceeding \(y\), is required in various problems in number theory. \textit{N. G. de Bruijn} [Indagationes Math. 13, 50-60 (1951; Zbl 0042.042)] showed that for all \(\epsilon >0\) and \(x>1\), \(y\geq 2\), \[ (1)\quad \psi(x,y)= \Lambda (x,y)+O_{\epsilon}(x \exp (-(\log y)^{3/5-\epsilon})), \] where \[ \Lambda(x,y)= x\int^{\infty}_{0}\rho (\frac{\log x-\log t}{\log y})d([t]/t),\quad for\quad x\not\in {\mathbb{N}} \] with \(\rho\) denoting the Dickman function, and he deduced the well known formula \[ \psi(x,y)= x \rho (u)(1+O_{\epsilon}(\log (u+1)/(\log y))),\quad u=(\log x)/(\log y), \] for a range which was eventually extended to \[ (2)\quad \exp ((\log \log x)^{5/3+\epsilon})\leq y\leq x,\quad x\geq x_ 0(\epsilon) \] by \textit{A. Hildebrand} [J. Number Theory 22, 289-307 (1986; Zbl 0575.10038)]. In Trans. Am. Math. Soc. 296, 265-290 (1986; Zbl 0601.10028) \textit{A. Hildebrand} and \textit{G. Tenenbaum} obtained another formula for \(\psi\) (x,y), in terms of \(\zeta(s,y)= \prod_{p\leq y}(1-p^{-s})^{-1}\) and other functions, which is valid whenever \(x\geq y\geq 2\). By building on these developments, the present author derives the following improvement of (1) in the range (2): \[ \psi(x,y)= \Lambda (x,y)(1+O_{\epsilon}(\exp (-(\log y)^{3/5-\epsilon}))). \]
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4106961
zbMATH Keywords: integers free of large prime factors
zbMATH Keywords: asymptotic formula
zbMATH Keywords: short interval estimate
zbMATH Keywords: Dickman function
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
cites work: The Turán-Kubilius inequality for integers without large prime factors.
cites work: On the size of prime factors of integers
cites work: Q5805143
cites work: Q5803354
cites work: On a problem of Oppenheim concerning ''Factorisatio Numerorum''
cites work: Q4071794
cites work: The Distribution of Round Numbers
cites work: Integers free of large prime factors and the Riemann hypothesis
cites work: On the number of positive integers \(\leq x\) and free of prime factors \(>y\)
cites work: On Integers Free of Large Prime Factors
cites work: On the number of prime factors of an integer
cites work: LOCAL DENSITIES OVER INTEGERS FREE OF LARGE PRIME FACTORS
cites work: APPLICATION OF SOME INTEGRAL EQUATIONS TO PROBLEMS OF NUMBER THEORY
cites work: Numbers with small prime factors, and the least 𝑘th power non-residue
cites work: Lois de répartition des diviseurs, 2
cites work: Sur un problème extrémal en arithmétique. (On an extremal problem in arithmetic)
cites work: Q5803461
Recommended article: Q3786539
Recommended article: Entiers Sans Grand Facteur Premier En Progressions Arithmetiques
Recommended article: On the number of prime factors of integers without large prime divisors
Recommended article: Sur le nombre des diviseurs premiers de n
Recommended article: An estimate for the number of integers without large prime factors
Recommended article: Q3679281
Recommended article: On the iteration of the number of divisors without large prime factors
Recommended article: Q3232350
Recommended article: On Integers Free of Large Prime Factors
description: scientific article; zbMATH DE number 4106961