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This is a Quantitative Study study.

October 16, 2025Open Access

Upper bounds on class numbers of real quadratic fields

RBRiccardo BernardiniUniversity of Udine

Key Points

The number of real quadratic fields with class number bound is at least x^{1/2-ε} for large x.
Class number estimates improve by a factor of log(log(d)) over a previous finding from 1971.
Algebraic conditions provide a lower bound for the size of fundamental units in real quadratic fields.
The proof corrects errors found in prior work on class numbers by Halter-Koch.

Abstract

We prove that, for any >0, the number of real quadratic fields Q (d) of discriminant d<x whose class number is d (d) ^-2 () ^-1 is at least x^1/2- for x large enough. This improves by a factor a result from 1971 by Yamamoto. We also establish a similar estimate for m-tuples of discriminants for any m 1. Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of Q (d), generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch.

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Riccardo Bernardini (Thu,) studied this question.

synapsesocial.com/papers/68f04acce559138a1a06e7e7 https://doi.org/https://doi.org/10.48550/arxiv.2506.21301

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