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

Abstract

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

arXiv comment26 pages, Number Tehory -- Diophantine Problems, Uniform Distribution and Applications, Festschrift in Honour of Robert F. Tichy's 60th Birthday, Springer 2017, ISBN 978-3-319-55356-6

References in corpus (1)

• Uniform distribution of subpolynomial functions along primes and applications

Cited by in corpus (1)

• On the probability of co-primality of two natural numbers chosen at random (Who was the first to pose and solve this problem?)