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

Weidong Gao; Jiangtao Peng; Qinghai Zhong

Weidong Gao ; Jiangtao Peng ; Qinghai Zhong

Acta Arithmetica, Tome 161 (2013), p. 271-285 / 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 $x {(l o g x)}^{1 - 1 / | G |} {(l o g l o g x)}^{_{k} (G)}$ . We prove, among other results, that $₁ (C_{n ₁} \oplus C_{n ₂}) = n ₁ + n ₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Publié le : 2013-01-01

Zbl 1290.11149

EUDML-ID : urn:eudml:doc:279710

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     title = {A quantitative aspect of non-unique factorizations: the Narkiewicz constants III},
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     volume = {161},
     year = {2013},
     pages = {271-285},
     zbl = {1290.11149},
     language = {en},
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Weidong Gao; Jiangtao Peng; Qinghai Zhong. A quantitative aspect of non-unique factorizations: the Narkiewicz constants III. Acta Arithmetica, Tome 161 (2013) pp. 271-285. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-3-6/