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On the distribution of ideals in cubic number fields

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Abstract

LetK be a cubic number field. Denote byA K (x) the number of ideals with ideal norm ≤x, and byQ K (x) the corresponding number of squarefree ideals. The following asymptotics are proved. For every ε>0 ε>0

$$\begin{gathered} {\text{ }}A_K (x) = c_1 x + O(x^{43/96 + \in } ), \hfill \\ Q_K (x) = c_2 x + O(x^{1/2} \exp {\text{ }}\{ - c(\log {\text{ }}x)^{3/5} (\log \log {\text{ }}x)^{ - 1/5} \} ). \hfill \\ \end{gathered}$$

Herec 1,c 2 andc are positive constants. Assuming the Riemann hypotheses for the Dedekind zeta function ζ K , the error term in the second result can be improved toO(x 53/116+ε).

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  1. Institut für Statistik, Technische Universität Graz, Lessingstrasse 27, A-8010, Graz, Austria

    Wolfgang Müller

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  1. Wolfgang Müller
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Müller, W. On the distribution of ideals in cubic number fields. Monatshefte für Mathematik 106, 211–219 (1988). https://doi.org/10.1007/BF01318682

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