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arXiv · 2610.28001

On the average of class numbers of indefinite binary quadratic forms

Yijie Diao
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Abstract

We prove that the sum of class numbers of primitive indefinite binary quadratic forms with integer coefficients over positive nonsquare discriminants less than $X$ is $O(X^{3/2}\sqrt{\log\log X}/(\log X)^{3/2})$. This improves the longstanding classical upper bound $O(X^{3/2}/\log X)$ and implies the same upper bound for the corresponding sum over real quadratic fields. Our main input is a uniform upper bound for the number of small fundamental solutions to Pell's equation. In particular, we prove that, for almost all nonsquare positive integers $D$, the fundamental solution $\varepsilon_D$ of $t^2-Du^2=1$ is larger than any fixed power of $D$.

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Yijie Diao. 2026-10-20. On the average of class numbers of indefinite binary quadratic forms. https://arxiv.org/abs/2610.28001

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