On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials

TL;DR

This paper investigates the density of natural numbers coprime to the floor of polynomial values, establishing asymptotic bounds and showing the density equals 1 over zeta(2) under certain conditions.

Contribution

It provides asymptotic bounds and proves the exact density for numbers coprime to the floor of polynomial values for specific polynomial classes.

Findings

Density of such naturals is exactly 1/ζ(2).

Asymptotic bounds are established for the count of these numbers.

Results depend on diophantine conditions on polynomial coefficients.

Abstract

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such naturals is exactly $\dfrac{1}{\zeta{\left(2\right)}}$.