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

Received: 23 July 2018

Revised: 29 August 2018

Accepted: 31 December 2018

Published: 24 October 2019

Correction: 31 July 2021