Counting principal ideals of small norm in the simplest cubic fields

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