Short interval results for squarefree numbers
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)] \(\vartheta =1057/4785=0.22089...\)
scientific article; zbMATH DE number 4144065
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