On the logarithmic probability that a random integral ideal is $\mathscr A$-free

Abstract

This extends a theorem of Davenport and Erdös 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 $\mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $\mathscr A$.

9 pages, to appear in S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk (eds.), Chowla's conjecture: from the Liouville function to the Möbius function, Lecture Notes in Math., Springer