The distribution of square-free numbers of the form \([n^c]\)
Let \(S_c(x)\) denote the number of positive integers \(n\) not exceeding \(x\) such that \([n^c]\) are square-free. The authors prove an asymptotic representation for \(S_c(x)\) with an explicit expression for the error term, provided that \(1< c< 2\). Now it is possible to use the method of exponential sums. It is shown that the sequence \([n^c]\) \((n=1,2,\dots)\) contains an infinite number of square-free integer whenever \(1< c<61/36\). This improves earlier results.
scientific article; zbMATH DE number 1307386
scientific article