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Burgess bounds for multi-dimensional short mixed character sums

Publication ,  Journal Article
Pierce, LB
Published in: Journal of Number Theory
June 1, 2016

This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in k≥ 1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation-dilation invariant system constructed in terms of f, we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions k≥ 2.

Duke Scholars

Published In

Journal of Number Theory

DOI

ISSN

0022-314X

Publication Date

June 1, 2016

Volume

163

Start / End Page

172 / 210

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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ICMJE
MLA
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Pierce, L. B. (2016). Burgess bounds for multi-dimensional short mixed character sums. Journal of Number Theory, 163, 172–210. https://doi.org/10.1016/j.jnt.2015.08.022
Pierce, L. B. “Burgess bounds for multi-dimensional short mixed character sums.” Journal of Number Theory 163 (June 1, 2016): 172–210. https://doi.org/10.1016/j.jnt.2015.08.022.
Pierce LB. Burgess bounds for multi-dimensional short mixed character sums. Journal of Number Theory. 2016 Jun 1;163:172–210.
Pierce, L. B. “Burgess bounds for multi-dimensional short mixed character sums.” Journal of Number Theory, vol. 163, June 2016, pp. 172–210. Scopus, doi:10.1016/j.jnt.2015.08.022.
Pierce LB. Burgess bounds for multi-dimensional short mixed character sums. Journal of Number Theory. 2016 Jun 1;163:172–210.
Journal cover image

Published In

Journal of Number Theory

DOI

ISSN

0022-314X

Publication Date

June 1, 2016

Volume

163

Start / End Page

172 / 210

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics