On the average of the number of imaginary quadratic fields with a given class number

TL;DR

This paper refines the asymptotic formula for the average count of imaginary quadratic fields with a given class number, improving error estimates and extending results to odd class numbers.

Contribution

It improves the error term in Soundararajan's asymptotic formula and extends the refinement to averages over odd class numbers.

Findings

Enhanced error bounds in asymptotic formulas

Refined average counts for imaginary quadratic fields

Extension of results to odd class numbers

Abstract

Let $\mathcal{F}(h)$ be the number of imaginary quadratic fields with class number $h$. In this note, we improve the error term in Soundararajan's asymptotic formula for the average of $\mathcal{F}(h)$. Our argument leads to a similar refinement of the asymptotic for the average of $\mathcal{F}(h)$ over odd $h$, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.