Counting principal ideals of small norm in the simplest cubic fields

Mikuláš Zindulka

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arXiv · 2501.06889

Counting principal ideals of small norm in the simplest cubic fields

Mikuláš Zindulka

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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.

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Mikuláš Zindulka. 2025-01-12. Counting principal ideals of small norm in the simplest cubic fields. https://arxiv.org/abs/2501.06889

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