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

Volume 158 / 2013

Weidong Gao, Jiangtao Peng, Qinghai Zhong Acta Arithmetica 158 (2013), 271-285 MSC: 11R27, 11B30, 11P70, 20K01. DOI: 10.4064/aa158-3-6

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 $x (\log x)^{1-1/|G|} (\log\log x)^{\mathsf N_k (G)}$. 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 $1< n_1\,|\,n_2$.

Authors

  • Weidong GaoCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
    e-mail
  • Jiangtao PengCollege of Science
    Civil Aviation University of China
    Tianjin 300300, P.R. China
    e-mail
  • Qinghai ZhongCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
    e-mail

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