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arXiv · 1709.00209

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

Wataru Takeda
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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 Lindelöf 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.

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Wataru Takeda. 2017-09-01. Uniform upper bounds of the distribution of relatively r-prime lattice points. https://arxiv.org/abs/1709.00209

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