On the average of class numbers of indefinite binary quadratic forms

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 $O(X^{3\mathrm{/}2}\sqrt{\log \log X}\mathrm{/}(\log X{)}^{3\mathrm{/}2})$. This improves the longstanding classical upper bound $O(X^{3\mathrm{/}2}\mathrm{/}\log X)$ 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 ${\epsilon }_D$ of $t^2-D{u}^2=1$ is larger than any fixed power of $D$.