Burgess bounds for short character sums evaluated at forms

Pierce, Lillian B.; Xu, Junyan

doi:10.2140/ant.2020.14.1911

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Burgess bounds for short character sums evaluated at forms

Lillian B. Pierce and Junyan Xu

Vol. 14 (2020), No. 7, 1911–1951

DOI: 10.2140/ant.2020.14.1911

Abstract

We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This $n$ -dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least $q^{β}$ , with $β > 1 ∕ 2 - 1 ∕ (2 (n + 1))$ . This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the second author.

Keywords

character sums

Mathematical Subject Classification 2010

Primary: 11L40

Milestones

Received: 17 July 2019

Revised: 14 December 2019

Accepted: 6 February 2020

Published: 18 August 2020

Authors

Lillian B. Pierce
Duke University Durham, NC United States

Junyan Xu
Indiana University Bloomington, IN United States

Vol. 14, No. 7, 2020