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

TL;DR

This paper generalizes a classical result about integers to the setting of number fields, providing a formula for the logarithmic density of ideals not divisible by any from a fixed set.

Contribution

It introduces a logarithmic density for sets of ideals in number fields and derives a formula for the density of $ ext{A}$-free ideals, extending Davenport and Erdős's theorem.

Findings

Derived a formula for the logarithmic density of $ ext{A}$-free ideals

Extended classical integer results to arbitrary number fields

Provided a framework for analyzing divisibility properties of ideals

Abstract

This extends a theorem of Davenport and Erd\"os 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 $\mathscr A$.