Upper bounds on class numbers of real quadratic fields

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