Complex moments of class numbers with fundamental unit restrictions

We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit ${\epsilon }_d$ is lower than $d^{1\mathrm{/}2+\alpha }$, where $0<\alpha <1\mathrm{/}2$. To do so we find an asymptotic formula for $z^{th}$-moments of such $h(d)$'s, over $d\le x$, uniformly for a complex number $z$ in a range of the form $\mathrm{\mid }z\mathrm{\mid }\le (\log x{)}^{1+o(1)}$, $\mathrm{\Re }(z)\ge -1$. This is achieved by constructing a probabilistic random model for these values, which we will use to obtain estimates for the distribution function of $h(d)$ over our family. As another application, we give an asymptotic formula for the number of $d$’s such that $h(d)\le H$ and ${\epsilon }_d\le d^{1\mathrm{/}2+\alpha }$ where $H$ is a large real number.