The ideal counting function in cubic fields

TL;DR

This paper investigates the behavior of the ideal counting function in cubic fields, providing an asymptotic formula for sums involving norms of ideals related to sums of two squares.

Contribution

It introduces an asymptotic formula for the sum of ideal counts over integers expressible as sums of two squares in cubic fields.

Findings

Derived an asymptotic estimate for the sum of ideal counts

Connected ideal counting to sums of two squares

Enhanced understanding of ideal distribution 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 $$ \sum\limits_{n_{1}^2+n_{2}^2\leq x}a_{K}(n_{1}^2+n_{2}^2). $$