Upper bounds on class numbers of real quadratic fields
Abstract
Abstract We prove that, for any $$\varepsilon >0$$ ε > 0 , the number of real quadratic fields $$\mathbb {Q}(\sqrt{d})$$ Q ( d ) of discriminant $$d d < x whose class number is $$\ll \sqrt{d}(\log {d})^{-2}(\log \log {d})^{-1}$$ ≪ d ( log d ) - 2 ( log log d ) - 1 is at least $$x^{1/2-\varepsilon }$$ x 1 / 2 - ε for x large enough. This improves by a factor $$\log \log {d}$$ log log d a result from 1971 by Yamamoto. We also establish a similar estimate for m -tuples of discriminants for any $$m\ge 1$$ m ≥ 1 . Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of $$\mathbb {Q}(\sqrt{d})$$ Q ( d ) , generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch.
Fields of Study & Subjects
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Limits and Structures in Graph Theory
Publication details
- Journal / source
- Monatshefte für Mathematik
- Publisher
- Springer Science and Business Media LLC
- Volume
- 210
- Issue
- 3-4
- Pages
- 371–392
- Publication date
- 13 July 2026
- Language
- EN