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

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 ≤ x. More generally, we obtain asymptotic formulas on the number of k-tuples (n₁,…,nₖ)∈ Nᵏ such that at least r pairs (nᵢ,nⱼ), 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).