Averages and moments associated to class numbers of imaginary quadratic fields

D. R. Heath-Brown, L. B. Pierce

Published 2014-09-10Version 1

For any odd prime $g$, let $h_g(-d)$ denote the $g$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $g=3$; nontrivial upper bounds for averages of $h_g(-d)$ have previously been known only for $g=3,5$. In this paper we prove nontrivial upper bounds for the average of $h_g(-d)$ for all primes $g \geq 7$, as well as nontrivial upper bounds for certain higher moments for all primes $g \geq 3$.