On the asymptotic densities of certain subsets of Nk
Received: 5 May 2001
Mathematics Subject Classification (2000): 11A25 - 11N37
Abstract: We determine the asymptotic density dk of the set of ordered k-tuples (n1,...,nk)∈Nk, k;≥ 2, such that there exists no prime power pa, a³≥ 1, appearing in the canonical factorization of each ni, 1 ≤ i ≤ k, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants dk and improve the error term of formula (1.2) due to 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 G. J. Rieger.