Short interval results for squarefree numbers
type: Item
instance of: scholarly article
title: Short interval results for squarefree numbers
zbMATH Open document ID: 0698.10027
DOI: 10.1016/0022-314X(90)90108-4
author: Michael Filaseta
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: Let \(\vartheta\) be the least upper bound of numbers \(\alpha\) for which all intervals \((x, x+x^{\alpha}]\) \((x\geq x_ 0(\alpha))\) contain a squarefree integer. The author describes two approaches to this classical problem. First he gives an elementary proof for \(\vartheta\leq 2/9\). This is \textit{H.-E. Richert}'s result, obtained by the use of exponential sums [J. Lond. Math. Soc. 29, 16-20 (1954; Zbl 0055.040)]. Using deeper facts on exponential sums, in particular the new exponent pairs by \textit{M. N. Huxley} and \textit{N. Watt} [Proc. Lond. Math. Soc., III. Ser. 57, 1-24 (1988; Zbl 0644.10027)] the author shows \(\vartheta \leq 47/217=0.21658...\) The best value hitherto was that of \textit{S. W. Graham} and \textit{G. Kolesnik} [Acta Arith. 49, 435-447 (1988; Zbl 0656.10041)] \(\vartheta =1057/4785=0.22089...\)
reviewed by: Dieter Wolke
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 4144065
zbMATH Keywords: short intervals
zbMATH Keywords: gaps between squarefree numbers
zbMATH Keywords: least upper bound
zbMATH Keywords: squarefree integer
zbMATH Keywords: exponential sums
zbMATH Keywords: exponent pairs
MaRDI profile type: Publication
cites work: Q4723831
cites work: An elementary approach to short interval results for \(k\)-free numbers
cites work: An Algorithm for Computing Optimal Exponent Pairs
cites work: On the difference between consecutive squarefree integers
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cites work: Q3735790
full work available at URL: 0022-314x(90)90108-4
OpenAlex ID: W1976148626
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description: scientific article; zbMATH DE number 4144065