ABSTRACT
Let K be an algebraic number field. In the 1960s and 1970s, P. Re´mond [351], W. Narkiewicz [319], [320] and J. S´liwa [381], [382] initiated the study of the asymptotic behavior of the following counting functions associated with non-unique factorizations:
M(x) = ∣∣{aOK | a ∈ A(OK) , (OK :aOK) ≤ x}∣∣ ,
Fk(x) = ∣∣{aOK | a ∈ O•K , (OK :aOK) ≤ x , |Z(a)| ≤ k}∣∣ and
Gk(x) = ∣∣{aOK | a ∈ O•K , (OK :aOK) ≤ x , |L(a)| ≤ k}∣∣ for every k ∈ N .
