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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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