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

Tom 124 / 2011

Weidong Gao, Yuanlin Li, Jiangtao Peng Colloquium Mathematicum 124 (2011), 205-218 MSC: Primary 11R27; Secondary 11P70, 20K01. DOI: 10.4064/cm124-2-5


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.


  • Weidong GaoCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
  • Yuanlin LiDepartment of Mathematics
    Brock University
    St. Catharines, Ontario, Canada L2S 3A1
  • Jiangtao PengCollege of Science
    Civil Aviation University of China
    Tianjin 300300, P.R. China

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