Halter-Koch, Franz Factorization problems in class number two. (English) Zbl 0816.11053 Colloq. Math. 65, No. 2, 255-265 (1993). Let \(R\) be the ring of integers of an algebraic number field \(K\) with class number \(h\neq 1\). It has been shown by the reviewer [Acta Arith. 21, 313-322 (1972; Zbl 0242.12007)] that the number of principal ideals \(I\) of \(R\) with \(N(I)\leq x\) which are generated by elements having at most \(k\) distinct factorizations into irreducibles is asymptotically equal to \(c_ k=c_ k x\log^ m x\log \log^{a_ k} x\) where \(m=- 1+1/h\), \(c_ k=c_ k (K)>0\) and \(a_ k\) is a positive integer, depending on \(k\) and the class group of \(K\). This has been generalized by the author and W. Müller [J. Reine Angew. Math. 421, 159-188 (1991; Zbl 0736.11064)] to the setting of abstract arithmetical formations. Now the author shows that if \(h=2\) then the exponent \(a_ k\) equals \(2n\), where \(n\) is the largest integer satisfying \(3\cdot 5\cdot 7\dots \cdot (2n-1) <k\) and gives an explicit description of \(c_ k\) in this case. Reviewer: W.Narkiewicz (Wrocław) Cited in 3 Documents MSC: 11R27 Units and factorization 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20M14 Commutative semigroups 11N37 Asymptotic results on arithmetic functions Keywords:class number two; factorizations; arithmetical formations Citations:Zbl 0242.12007; Zbl 0736.11064 PDFBibTeX XMLCite \textit{F. Halter-Koch}, Colloq. Math. 65, No. 2, 255--265 (1993; Zbl 0816.11053) Full Text: DOI EuDML