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
Received: 23 July 2018
Revised: 29 August 2018
Accepted: 31 December 2018
Published: 24 October 2019
Authors
Wataru Takeda
Department of Mathematics
Nagoya University
Chikusa-ku
Nagoya
Japan