On Moments of non-normal number fields

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.