The distribution of class numbers in a special family of real quadratic fields

TL;DR

This paper studies the distribution of class numbers in a specific family of real quadratic fields, revealing similarities with imaginary quadratic fields and deriving average order formulas for class numbers.

Contribution

It introduces and analyzes Chowla's family of real quadratic fields, establishing distributional properties and average class number formulas, extending to Yokoi's family with minor modifications.

Findings

Distribution of class numbers in Chowla's family resembles that of imaginary quadratic fields.

Average order of fields with class number h is approximately (log h)/2G.

Results extend to Yokoi's family with minor adjustments.

Abstract

We investigate the distribution of class numbers in the family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ corresponding to fundamental discriminants of the form $d=4m^2+1$, which we refer to as Chowla's family. Our results show a strong similarity between the distribution of class numbers in this family and that of class numbers of imaginary quadratic fields. As an application of our results, we prove that the average order of the number of quadratic fields in Chowla's family with class number $h$ is $(\log h)/2G$, where $G$ is Catalan's constant. With minor modifications, one can obtain similar results for Yokoi's family of real quadratic fields $\mathbb{Q}(\sqrt{d})$, which correspond to fundamental discriminants of the form $d=m^2+4$.