Averages and moments associated to class numbers of imaginary quadratic fields

TL;DR

This paper establishes new upper bounds for the averages and higher moments of the -part of class numbers of imaginary quadratic fields for all primes  , extending previous results limited to smaller primes.

Contribution

It provides the first nontrivial upper bounds for averages of $h_(-d)$ for all primes   and for higher moments for all primes  , advancing understanding of class number distributions.

Findings

Established upper bounds for averages of $h_(-d)$ for all primes  .

Derived bounds for higher moments of $h_(-d)$ for all primes  .

Extended previous results limited to =3,5 to all primes  .

Abstract

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