Higher moments related to Dedekind zeta functions of non-normal fields

Abstract

Let K be a non-normal number field over $Q$ with $a_K\left(m\right)$ the number of integer ideals in K of norm $m\in Z$. Let L be the Galois closure of K and assume that $\mathrm{Gal}(L/K)$ is monomial. We obtain an asymptotic formula for the summation ${\sum }_{m\le x}{a}_K{\left(m\right)}^l$ for any $l\ge 1$. Moreover, in the dihedral case, we also obtain asymptotic formulas for the summation over a binary quadratic form. If K is a non-normal cubic field, this work refines previous works.