Counting principal ideals of small norm in the simplest cubic fields

TL;DR

This paper estimates the count of principal ideals with small norm in simplest cubic fields, revealing many such ideals exist even for small norms, and provides accurate order of magnitude estimates across a wide range of x.

Contribution

It offers the first precise order of magnitude estimates for principal ideals of small norm in simplest cubic fields, valid for all x ≥ 1.

Findings

Many principal ideals of small norm exist in simplest cubic fields.

The estimate holds even when x is much smaller than the discriminant.

Provides the correct order of magnitude for the count of such ideals.

Abstract

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.