We prove that, for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the number of real quadratic fields <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{d})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>d</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of discriminant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d<x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo><</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> whose class number is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ll \sqrt{d}(\log {d})^{-2}(\log \log {d})^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≪</mo> <msqrt> <mi>d</mi> </msqrt> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mo>log</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is at least <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x^{1/2-\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>-</mo> <mi>ε</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> for x large enough. This improves by a factor <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\log \log {d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo>log</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> a result from 1971 by Yamamoto. We also establish a similar estimate for m-tuples of discriminants for any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$m\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{d})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>d</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch.

Upper bounds on class numbers of real quadratic fields

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2026年7月13日

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Riccardo Bernardini

摘要

We prove that, for any $\varepsilon >0$ $ε > 0$ , the number of real quadratic fields $\mathbb {Q}(\sqrt{d})$ $Q (\sqrt{d})$ of discriminant $d<x$ $d < x$ whose class number is $\ll \sqrt{d}(\log {d})^{-2}(\log \log {d})^{-1}$ $≪ \sqrt{d} {(log d)}^{- 2} {(log log d)}^{- 1}$ 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 $m\ge 1$ $m \geq 1$ . Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of $\mathbb {Q}(\sqrt{d})$ $Q (\sqrt{d})$ , generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch.