On the probability that the integer parts of polynomials are coprime

Abstract

We prove that for any $s\ge 2$ and polynomials $f_1(x),\dots ,f_{s-1}(x)\in \mathbb{R}[x]$ satisfying a certain equidistribution condition, the natural density of the numbers $n\in \mathbb{N}$ such that $n,\lfloor f_1(n)\rfloor ,\dots ,\lfloor f_{s-1}(n)\rfloor$ are mutually coprime is $1/\zeta (s)$. This generalises a recent result of Champagne, in which this was proven under the assumption that $f_1$ is linear.

arXiv comment26 pages