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

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Let K be an algebraic number field with non-trivial class group G and OK be its ring of integers. For k 2 N and some real x 1, let Fk(x) denote the number of non-zero principal ideals aOK with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk(x) behaves, for x ! 1, asymptotically like x(logx) 1/|G| 1 (loglogx) Nk(G) . In this article, it is proved that for every prime p, N1(Cp Cp) = 2p, and it is also proved that N1(Cmp Cmp) = 2mp if N1(Cm Cm) = 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(Cmn Cmn) = 2mn. Our results confirm a conjecture given by W. Narkiewicz thirty years ago partly and improve the known results substantially.