We prove that, for any >0, the number of real quadratic fields Q (d) of discriminant d<x whose class number is d (d) ^-2 () ^-1 is at least x^1/2- for x large enough. This improves by a factor 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.
Riccardo Bernardini (Thu,) studied this question.