Distribution of class numbers in continued fraction families of real quadratic fields

Abstract

We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2, \dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ and generalize well-known families such as Chowla's $4n^2+1$, for which analogous results were recently proved by Dahl and Lamzouri.