The distribution of $k$-free ideals in ray class groups

Abstract

In this paper, we extend the classical problem of studying the distribution of $k$-free integers in arithmetic progressions to the setting of arbitrary number fields. Using the language of ray class groups, we establish asymptotic formulas, together with error terms, for the number of $k$-free ideals of bounded norm lying in a given ray class. In particular, our results show that $k$-free ideals are equidistributed among ray classes. We also obtain improved error estimates in the cases of ideal class groups and narrow class groups by using sharper ideal counting asymptotics due to Landau. Our results recover the classical formulas of Gegenbauer and Cohen--Robinson over $\mathbb{Q}$ and extend previous work of Benkowski, Nymann, and Sittinger to the setting of ray class groups. We also present explicit computational examples that illustrate the asymptotic formulas and the equidistribution of $k$-free ideals among ray classes.