Relatively prime pairs in the Piatetski-Shapiro sequences

Abstract

In this paper, we prove an asymptotic formula for the number of relatively prime pairs in the Piatetski-Shapiro sequence of arbitrarily large order. This improves the result of Pimsert, Srichan and Tangsupphathawat (2023), the order of which was restricted to be $<\frac{3}{2}$. The key ingredients of the proof are a simple averaging trick and an extension of Deshouillers' result on the distribution of the Piatetski-Shapiro sequence in arithmetic progressions.