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

Weidong Gao ¹   ; Yuanlin Li ²   ; Jiangtao Peng ³

¹ Center for Combinatorics Nankai University LPMC-TJKLC Tianjin 300071, P.R. China
² Department of Mathematics Brock University St. Catharines, Ontario, Canada L2S 3A1
³ College of Science Civil Aviation University of China Tianjin 300300, P.R. China

Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 205-218

Résumé

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.

Détail

DOI : 10.4064/cm124-2-5

Keywords: algebraic number field non trivial class group mathcal its ring integers mathbb real denote number non zero principal ideals mathcal norm bounded has distinct factorizations irreducible elements known behaves infty asymptotically log log log mathsf article proved every prime mathsf oplus proved mathsf oplus mathsf oplus large enough particular shown each positive integer there positive integer mathsf oplus results partly confirm conjecture given narkiewicz thirty years ago improve known results substantially