The Piltz divisor Problem in Number Fields Using The Resonance Method

TL;DR

This paper extends the Piltz divisor problem to number fields, providing improved bounds for error terms by generalizing a Voronoi formula and applying resonance methods.

Contribution

It introduces a novel approach by generalizing a Voronoi-type formula to number fields and applying resonance techniques to improve error bounds.

Findings

Obtained improved Omega bounds for the divisor problem in number fields.

Generalized Voronoi formula to the setting of number fields.

Applied resonance method to achieve sharper error estimates.

Abstract

The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved $\Omega-$bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.