Upper bounds on class numbers of real quadratic fields

We prove that, for any ε>0, the number of real quadratic fields Q(√d) of discriminant $d<x$ whose class number is √d(logd)⁻²(loglogd)⁻¹ is at least x1/2-ε for x large enough. This improves by a factor loglogd 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.