### A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. We prove, among other results, that $\u2081\left({C}_{n\u2081}\oplus {C}_{n\u2082}\right)=n\u2081+n\u2082$ for all integers n₁,n₂ with 1 < n₁|n₂.