Double character sums over subgroups and intervals

Mei Chu Chang, Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

We estimate double sums Sχ(a, I, G) = ∑x∈Iλ∈Gχ(x + aλ), 1 ≤ a < p - 1, with a multiplicative character χ modulo p where I = {1, . . . , H} and G is a subgroup of order T of the multiplicative group of the finite field of p elements. A nontrivial upper bound on S χ (a, I, G) can be derived from the Burgess bound if H ≥ p1/4+ε and from some standard elementary arguments if T ≥ p1/2+ε, where ε > 0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters H and T . We also estimate double sums Tχ(a, G) = ∑χ,μ∈G χ(a + λ + μ), 1 ≤ a < p - 1, and give an application to primitive roots modulo p with three nonzero binary digits.

Original languageEnglish
Pages (from-to)376-390
Number of pages15
JournalBulletin of the Australian Mathematical Society
Volume90
Issue number3
DOIs
Publication statusPublished - 2 Dec 2014
Externally publishedYes

Keywords

  • Character sums
  • Intervals
  • Multiplicative subgroups of finite fields

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