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

Weidong Gao ; Yuanlin Li ; Jiangtao Peng

Colloquium Mathematicae, Tome 122 (2011), p. 205-218 / Harvested from The Polish Digital Mathematics Library

Résumé

Let K be an algebraic number field with non-trivial class group G and $_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_{k}(x)$ denote the number of non-zero principal ideals $a_{K}$ 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 → ∞, asymptotically like $x(log x)^{1/|G|-1} (loglogx)^{_{k}(G)}$. In this article, it is proved that for every prime p, $₁(C_{p}⊕ C_{p}) = 2p$, and it is also proved that $₁ (C_{mp}⊕ C_{mp}) = 2mp$ if $₁ (C_{m}⊕ C_{m}) = 2m$ and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that $₁(C_{mn}⊕ C_{mn}) = 2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.

Détail

Publié le : 2011-01-01

Zbl 1246.11173

EUDML-ID : urn:eudml:doc:284341