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

### Volume 124 / 2011

#### Abstract

Let $K$ be an algebraic number field with non-trivial class group $G$ and $\mathcal O_K$ be its ring of integers. For $k \in \mathbb N$ and some real $x \ge 1$, let $F_k (x)$ denote the number of non-zero principal ideals $a\mathcal O_K$ with norm bounded by $x$ such that $a$ has at most $k$ distinct factorizations into irreducible elements. It is well known that $F_k (x)$ behaves, for $x \to \infty$, asymptotically like $x (\log x)^{1/|G|-1} (\log\log x)^{\mathsf N_k (G)}$. In this article, it is proved that for every prime $p$, $\mathsf N_1 (C_p\oplus C_p)=2p$, and it is also proved that $\mathsf N_1 (C_{mp}\oplus C_{mp})=2mp$ if $\mathsf N_1 (C_m\oplus C_m)=2m$ and $m$ is large enough. In particular, it is shown that for each positive integer $n$ there is a positive integer $m$ such that $\mathsf N_1(C_{mn}\oplus C_{mn})=2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.