The number of imaginary quadratic fields with a given class number

We investigate the number $\text{Cal}F(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of $\text{Cal}F(h)$. 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 $\text{Cal}F(h)$.