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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.

##### 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
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