On the density of coprime tuples of the form $(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)$, where $f_1,\ldots,f_k$ are functions from a Hardy field

TL;DR

This paper investigates the density of tuples formed by an integer and the floors of functions from a Hardy field, establishing conditions under which the density of coprime tuples exists and equals a specific value involving the Riemann zeta function.

Contribution

It proves the existence and exact value of the density of coprime tuples involving Hardy field functions under natural conditions.

Findings

Density of coprime tuples exists under certain conditions.

The density equals 1 divided by the Riemann zeta function at (k+1).

Provides a link between Hardy field functions and number-theoretic density results.

Abstract

Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)=1\big\} $$ exists and equals $\frac{1}{\zeta(k+1)}$, where $\zeta$ is the Riemann zeta function.