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Factorization problems in class number two. (English) Zbl 0816.11053
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.

11R27 Units and factorization
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20M14 Commutative semigroups
11N37 Asymptotic results on arithmetic functions
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