Burgess bounds for short character sums evaluated at forms

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 $q^{\beta }$, with $\beta >1/2-1/(2(n+1))$. 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.

28 pages

References in corpus (9)

• The Weyl bound for Dirichlet $L$-functions of cube-free conductor
• Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli
• The Burgess bound via a trivial delta method
• Burgess bounds for short mixed character sums
• Small Solutions of Quadratic Congruences, and Character Sums with Binary Quadratic Forms
• Character sums to smooth moduli are small
• Some mixed character sums
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• The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture

Cited by in corpus (1)

• Burgess bounds for short character sums evaluated at forms II: the mixed case