  We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on
the number of $k$-compositions of $n$ satisfying some coprimality conditions.
We substantially refine the error term concerning the number of
$k$-compositions of $n$ with pairwise relatively prime summands. We use a
different approach, based on properties of multiplicative arithmetic functions
of $k$ variables and on an asymptotic formula for the restricted partition
function.
