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We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) onthe 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 adifferent approach, based on properties of multiplicative arithmetic functionsof $k$ variables and on an asymptotic formula for the restricted partitionfunction.

On the number of $k$-compositions of $n$ satisfying certain coprimality conditions