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On the Logarithmic Probability That a Random Integral Ideal Is \( \pmb{\mathcal A}\)-Free

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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

Abstract

This extends a theorem of Davenport and Erdös (J Indian Math Soc 15:19–24, 1951) on sequences of rational integers to sequences of integral ideals in arbitrary number fields K. More precisely, we introduce a logarithmic density for sets of integral ideals in K and provide a formula for the logarithmic density of the set of so-called \(\mathbb A\)-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set \(\mathbb A\).

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Acknowledgements

It is a pleasure to thank Joanna Kułaga-Przymus and Jeanine Van Order for helpful discussions. This work was initiated during a research in pairs stay at CIRM (Luminy) in 2016, within the Jean Morlet Chair. It was supported by the German Research Council (DFG), within the CRC 701.

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Correspondence to Christian Huck .

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Huck, C. (2018). On the Logarithmic Probability That a Random Integral Ideal Is \( \pmb{\mathcal A}\)-Free. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_13

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