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Ein quantitatives Resultat über Faktorisierungen verschiedener Länge in algebraischen Zahlkörpern. (A quantitative result on factorizations of different length in algebraic number fields). (German) Zbl 0721.11041
Let R denote the ring of algebraic integers in an algebraic number field K with the class group G of order $$h\geq 3$$. For a natural $$m\geq 1$$ and real $$x\geq 1$$ let $$G_ m(x)$$ $$(\bar G_ m(x))$$ denote the number of principal ideals aR such that $$N(aR)\leq x$$ and a has at most m (exactly m resp.) factorizations into irreducibles of distinct lengths. It is known that $G_ m(x)=(C+o(1))x(\log x)^{-\eta (G,m)}\quad (\log \log x)^{\psi (G,m)},$
$\bar G_ m(x)=(\bar C+o(1))x(\log x)^{-{\bar \eta}(G,m)}\quad (\log \log x)^{{\bar \psi}(G,m)},$ the constants $$C$$, $$\bar C$$, $$\eta(G,m)$$, $${\bar \eta}(G,m)$$, $$\psi(G,m)$$ and $${\bar\psi}(G,m)$$ being positive. The author’s main results give explicit (combinatorial) formulae for the exponents in the above asymptotics.

##### MSC:
 11R27 Units and factorization 11N45 Asymptotic results on counting functions for algebraic and topological structures
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##### References:
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