Some multiplicative properties of products of natural numbers in a short interval
type: Item
instance of: scholarly article
title: Some multiplicative properties of products of natural numbers in a short interval
zbMATH Open document ID: 0846.11053
author: N. M. Khodzhaev
published in: Moscow University Mathematics Bulletin
publication date: 1996-03-27T00:00:00Z
review text: Let \(\varepsilon > 0\) be a small real number and \(x > x_0 (\varepsilon)\). The author proves that the interval \((x,x + zx^\varepsilon]\) contains many \(z\)-smooth integers, that is integers, all of whose prime factors are at most \(z\), where \(z \geq x^{11/26 + \varepsilon}\). The result is based on a deep estimate of \textit{H. Iwaniec} and \textit{C. J. Mozzochi} [J. Number Theory 29, 60-93 (1988; Zbl 0644.10031)]. The above result has various consequences of the following type. Let \(k(n)\) denote the smallest positive integer \(k\) such that the real numbers \(\log (n + 1), \dots, \log (n + k)\) are linearly dependent over the rational numbers. We have \(k(n) \leq n^{11/26 + \varepsilon}\).
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 848930
zbMATH Keywords: prime divisors of integers in short intervals
zbMATH Keywords: products of integers in short intervals
zbMATH Keywords: smooth integers
reviewed by: Antal Balog
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 848930