#### Vol. 301, No. 2, 2019

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Uniform bounds of the Piltz divisor problem over number fields

### Wataru Takeda

Vol. 301 (2019), No. 2, 601–616
DOI: 10.2140/pjm.2019.301.601
##### Abstract

We consider the upper bound of the Piltz divisor problem over number fields. The 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 the ideal counting function and relatively $r$-prime lattice points as one of its applications.

##### Keywords
ideal counting function, exponential sum, Piltz divisor problem
##### Mathematical Subject Classification 2010
Primary: 11N45
Secondary: 11R42, 11H06, 11P21
##### Milestones
Revised: 29 August 2018
Accepted: 31 December 2018
Published: 24 October 2019
##### Authors
 Wataru Takeda Department of Mathematics Nagoya University Chikusa-ku Nagoya Japan