The least common multiple of sets of positive integers

It is well known that the classical Chebyshev's function $\psi(n)=\sum_{m<n}\Lambda(m)$ has an alternative expression in terms of the least common multiple of the first n integers:$\psi(n)=\text{log lcm} (1,2,\dots, n)$.

Here we generalize this function by considering, for a set $\mathcal A\subseteq  [1,n]$, the quantity $\psi(\mathcal A):=\text{log lcm} \{a\,:\, a\in\mathcal A\}$ and we ask ourselves about its asymptotic behavior.

We will focus on sets given by $\mathcal A_f=\{ f(1), f(2), ..., f(n)\}$ for some polynomial with integer coefficients. We will also discuss the case where the set is chosen at random in $[1,n]$ with prescribed size.