On Moments of non-normal number fields

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.