On the Density of Coprime Tuples of the Form (n, ⌊f1(n)⌋, …, ⌊fk(n)⌋), Where f_1, …, f_k Are Functions from a Hardy Field

Let k∈Nk∈Nk \in \mathbb{N} and let f1, …, f k belong to a Hardy field. We prove that under some natural conditions on the k-tuple ( f1, …, f k ) the density of the set {n∈N:gcd(n,⌊f1(n)⌋,…,⌊fk(n)⌋)=1}{n∈N:gcd(n,⌊f1(n)⌋,…,⌊fk(n)⌋)=1}\displaystyle{\big{n \in \mathbb{N}:\gcd (n,\lfloor ,f_{1}(n)\rfloor,\ldots,\lfloor ,f_{k}(n)\rfloor ) = 1\big}} exists and equals 1ζ(k+1)1ζ(k+1)\frac{1} {\zeta (k+1)}, where ζ is the Riemann zeta function.