The ideal counting function in cubic fields

Abstract

For a cubic algebraic extension $K$ of $\mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_K(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$\sum _{n_1^2+n_2^2\le x}{a}_K(n_1^2+n_2^2).$$