Rough values of Piatetski-Shapiro sequences
The author extends investigation of Piatetski-Shapiro sequences \(\{\lfloor n^c\rfloor\}_{n\in\mathbb N}\) by considering distribution of \(n\) with \(\lfloor n^c\rfloor\) free of small primes. More precisely, an integer is said to be \(y\)-rough if all of its prime factors are greater than \(y\), and the object here is an asymptotic formula for the counting function \[ \Phi_c(x,y)=\#\{\, n\leq x:\lfloor n^c\rfloor \text{ is y-rough}\,\}, \] as \(x\to\infty\). The author succeeds in his task for \(c\) in the range \(1<c<\frac{2229}{1949}\), and certain corresponding ranges of \(y\). The result is to complicated to state here explicitly. By way of comparison, \(\frac{2229}{1949}=1.143666...\), and an asymptotic formula for the number of primes not exceeding \(x\) of the form \(\lfloor n^c\rfloor\) is known to hold for \(1<c<\frac{2817}{2426}=1.16117...\). The latter result is due to \textit{J. Rivat} and \textit{P. Sargos} [Can. J. Math. 53, No. 2, 414--433 (2001; Zbl 0970.11035)].
scientific article; zbMATH DE number 6830184