LASZLO TOTH

On the asymptotic densities of certain subsets of N^k

Pages: 121-131
Received: 5 May 2001   
Mathematics Subject Classification (2000): 11A25 - 11N37

Abstract: We determine the asymptotic density d_k of the set of ordered k-tuples (n₁,...,n_k)∈N^k, k;≥ 2, such that there exists no prime power p^a, a³≥ 1, appearing in the canonical factorization of each n_i, 1 ≤ i ≤ k, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants d_k 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.