Complex moments of class numbers with fundamental unit restrictions

TL;DR

This paper studies the distribution of class numbers of indefinite binary quadratic forms with restrictions on the fundamental unit, providing asymptotic formulas for moments and counts within this family.

Contribution

It introduces a probabilistic model to analyze class number moments under fundamental unit restrictions, deriving new asymptotic formulas and distribution estimates.

Findings

Asymptotic formula for moments of class numbers with fundamental unit bounds

Distribution estimates for class numbers in the specified family

Asymptotic count of discriminants with bounded class number and fundamental unit

Abstract

We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit $\varepsilon_d$ is lower than $d^{1/2+\alpha}$, where $0<\alpha<1/2$. To do so we find an asymptotic formula for $z^{th}$-moments of such $h(d)$'s, over $d\leq x$, uniformly for a complex number $z$ in a range of the form $|z|\leq(\log x)^{1+o(1)}$, $\Re(z)\geq -1$. 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 $h(d)\leq H$ and $\varepsilon_d\leq d^{1/2+\alpha}$ where $H$ is a large real number.