On Moments of non-normal number fields

TL;DR

This paper derives asymptotic formulas for sums of powers of ideal counting functions in number fields, extending previous results to non-normal fields and higher powers, thus broadening the understanding of ideal distributions.

Contribution

It generalizes existing asymptotic formulas for ideal counts to non-normal number fields and higher powers, unifying previous special cases.

Findings

Derived asymptotic formulas for sums of $a_K(m)^l$ in general number fields.

Extended known results from Galois and cubic fields to broader classes.

Provided a unified approach covering previous special cases.

Abstract

Let $ K $ be a number field over $ \mathbb{Q} $ and let $ a_K(m) $ denote the number of integral ideals of $ K $ of norm equal to $ m\in\mathbb{N} $. In this paper we obtain asymptotic formulae for sums of the form $ \sum_{m\leq X} a^l_K(m) $ thereby generalizing the previous works on the problem. Previously such asymptotics were known only in the case when $ K $ is Galois or when $K$ was a non normal cubic extension and $ l=2,3 $. The present work subsumes both these cases.