On the asymptotic densities of certain subsets of $\bf N^k$

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants $\delta_k$ and improve the error term of formula (1.2) due to {\sc E. Cohen}. We point out that our treatment, based on certain inversion functions, is applicable also in case $k=1$ in order to establish asymptotic formulae with error terms regarding the densities of subsets of $\N$ with additional multiplicative properties. These generalize an often cited result of {\sc G. J. Rieger}.