On the asymptotic density of $k$-tuples of positive integers with pairwise non-coprime components

Abstract

We use the convolution method for arithmetic functions of several variables to deduce an asymptotic formula for the number of $k$-tuples of positive integers with components which are pairwise non-coprime and $\le x$. More generally, we obtain asymptotic formulas on the number of $k$-tuples $(n_1,\ldots,n_k)\in {\Bbb N}^k$ such that at least $r$ pairs $(n_i,n_j)$, respectively exactly $r$ pairs are coprime. Our results answer the questions raised by Moree (2005, 2014), and generalize and refine related results obtained by Heyman (2014) and Hu (2014).