  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.
