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

Weidong Gao¹ ; Jiangtao Peng² ; Qinghai Zhong¹

¹ Center for Combinatorics Nankai University Tianjin 300071, P.R. China
² College of Science Civil Aviation University of China Tianjin 300300, P.R. China

Acta Arithmetica, Tome 158 (2013) no. 3, pp. 271-285

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 $x (\log x)^{1-1/|G|} (\log\log x)^{\mathsf N_k (G)}$. We prove, among other results, that $\mathsf N_1 (C_{n_1}\oplus C_{n_2})=n_1+n_2$ for all integers $n_1,n_2$ with $1 n_1\,|\,n_2$.

Détail

DOI : 10.4064/aa158-3-6

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 prove among other results mathsf oplus integers