Uniform bounds of Piltz divisor problem over number fields

TL;DR

This paper improves the error bounds for the Piltz divisor problem over number fields, extending classical results and providing uniform estimates for related counting functions using exponential sum techniques.

Contribution

It introduces new bounds for the Piltz divisor problem over number fields and establishes uniform estimates for ideal counting and lattice points.

Findings

Improved error bounds for Piltz divisor problem over number fields.

Established uniform estimates for ideal counting functions.

Applied exponential sum estimates to achieve these results.

Abstract

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.