Double character sums over subgroups and intervals

Authors

  • M. C. Chang Department of Mathematics, University of California, Riverside
  • I. E. Shparlinksi UNSW

Keywords:

character sums, intervals, multiplicative subgroups of finite fields

Abstract

We estimate double sums Sχ(a,I,G)=xIλGχ(x+aλ),1a<p1, 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 Sχ(a,I,G) can be derived from the Burgess bound if Hp1/4+ε and from some standard elementary arguments if Tp1/2+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)=λ,μGχ(a+λ+μ),1a<p1, and give an application to primitive roots modulo p with 3 non-zero binary digits. DOI:- 10.1017/S0004972714000227

Published

2014-09-20

Issue

Section

Articles