The distribution of smooth numbers and ideals in rings of integers of number fields

Abstract

In this paper, for functions g(x), we evaluate the asymptotic behavior (as x → ∞) of the proportion of ideals of the ring of integers of some number field K that are g(x)-smooth. For certain rates of growth of g(x), there are established proofs that give such asymptotic behavior of |T(x, g(x))|, where T(x, g(x)) is defined as the set of g(x)-smooth integers less than x, within a factor of 1+o(1). Ennola’s proof covers the case in which, asymptotically in x, g(x) ≤ (ln x)3/4, and Hildebrand’s covers the case where g(x) ≥ x (ln ln x) 5/3+ϵ/(ln x). In this paper, we give proofs of slightly modified versions of these results, and show that they can be easily generalized to smooth integers in number fields. We also show that smooth ideals in the ring of integers of a number field are asymptotically evenly distributed across all ray classes for any ray class group, which further allows us to extend Buchstab’s result (that smooth numbers are evenly distributed across modular congruence classes) to principal ideal domains.