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 ${\sum }_{m\le X}{a}_K^l(m)$ 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.

13 Pages