The number of imaginary quadratic fields with a given class number

TL;DR

This paper studies the distribution of imaginary quadratic fields with a fixed class number, providing asymptotic formulas, bounds, and applications to related number theory questions.

Contribution

It establishes an asymptotic formula for the average number of such fields and provides a new upper bound for their count.

Findings

Derived an asymptotic formula for the average of ${\Cal F}(h)$

Established a non-trivial upper bound for ${\Cal F}(h)$

Applied results to questions on the odd part of class numbers

Abstract

We investigate the number ${\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\Cal F}(h)$ and give an application to a question of Rosen and Silverman on the odd part of the class number. Finally, we speculate on the asymptotic nature of ${\Cal F}(h)$.