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

• Autorzy: Weidong Gao , Yuanlin Li , Jiangtao Peng
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11P70: Inverse problems of additive number theory, including sumsets
• 11R27: Units and factorization
• 20K01: Finite abelian groups [For sumsets, see and ]

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2011
• Tom: 124
• Numer: 2
• Strony: 205-218

Daty

wydano

2011

Twórcy

autor

Weidong Gao

• Center for Combinatorics, Nankai University, LPMC-TJKLC, Tianjin 300071, P.R. China

autor

Yuanlin Li

• Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1

autor

Jiangtao Peng

• College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China

Identyfikatory

DOI

10.4064/cm124-2-5