Uniform upper bounds of the distribution of relatively r-prime lattice points

TL;DR

This paper improves uniform upper bounds on the distribution of relatively r-prime lattice points in number fields, removing previous assumptions and providing better bounds for cubic extensions.

Contribution

It removes all assumptions about number fields and enhances uniform upper bounds for the distribution of lattice points, especially in cubic extension fields.

Findings

Unconditional uniform upper bounds for lattice points in number fields.

Improved bounds for cubic extension fields.

Better bounds than those assuming Lindelöf Hypothesis.

Abstract

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under assuming the Lindel\"of hypothesis. And we also showed unconditional results for abelian extensions with a degree less than or equal to $6$. In this paper we remove all assumption about number fields and improve uniform upper bounds. Throughout this paper we consider estimates for distribution of ideals of the ring of integer $\mathcal{O}_K$ and obtain uniform upper bounds. And when $K$ runs through cubic extension fields we show better uniform upper bounds than that under the Lindel\" of Hypothesis.