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arXiv · 1907.03108

Burgess bounds for short character sums evaluated at forms

Lillian B. Pierce·Junyan Xu
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Abstract

In this work 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.

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Lillian B. Pierce, Junyan Xu. 2019-07-06. Burgess bounds for short character sums evaluated at forms. https://doi.org/10.2140/ant.2020.14.1911

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