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2011 | 124 | 2 | 205-218
Tytuł artykułu

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

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Let K be an algebraic number field with non-trivial class group G and $𝓞_{K}$ 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 $a𝓞_{K}$ 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 $x(log x)^{1/|G|-1} (loglogx)^{𝖭_{k}(G)}$. In this article, it is proved that for every prime p, $𝖭₁(C_{p}⊕ C_{p}) = 2p$, and it is also proved that $𝖭₁ (C_{mp}⊕ C_{mp}) = 2mp$ if $𝖭₁ (C_{m}⊕ C_{m}) = 2m$ and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that $𝖭₁(C_{mn}⊕ C_{mn}) = 2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.
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  • Center for Combinatorics, Nankai University, LPMC-TJKLC, Tianjin 300071, P.R. China
autor
  • Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1
  • College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-2-5
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