zbMATH — the first resource for mathematics

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.

11R27 Units and factorization
11N45 Asymptotic results on counting functions for algebraic and topological structures
Full Text: DOI EuDML
[1] Geroldinger, A.: Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z.197, 505–529 (1988) · Zbl 0618.12002
[2] Geroldinger, A.: On non-unique factorizations into irreducible elements II. Coll. Math. Soc. J. Bolyai, 51. Number Theory, Budapest, pp. 723–757 (1987) · Zbl 0619.13009
[3] Kaczorowski, J.: Some remarks on factorization in algebraic number fields. Acta Arith.43, 53–68 (1983) · Zbl 0526.12006
[4] Narkiewicz, W.: On algebraic number fields with non-unique factorization. Colloq. Math.12, 59–68 (1964) · Zbl 0127.26503
[5] Narkiewicz, W.: Elementary and analytic theory of algebraic numbers. Warszawa: PWN 1974 · Zbl 0276.12002
[6] Narkiewicz, W.: Finite abelian groups and factorization problems. Colloq. Math.42, 319–330 (1979) · Zbl 0514.12004
[7] Skula, L.: Onc-semigroups. Acta Arith.31, 247–257 (1976) · Zbl 0303.13014
[8] Śliwa, J.: Factorizations of distinct lengths in algebraic number fields. Acta Arith.31, 399–417 (1976) · Zbl 0347.12005
[9] Śliwa, J.: Remarks on factorizations in algebraic number fields. Colloq. Math.46, 123–130 (1982) · Zbl 0514.12005
[10] Zaks, A.: Half-factorial domains. Bull. Am. Math. Soc.82, 721–723 (1976) · Zbl 0338.13020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.