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 \( S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1, \) with a multiplicative character \(\chi\) modulo \(p\) where \(I= \{1,\ldots, H\}\) and \(G\) is a subgroup of order \(T\) of the multiplicative group of the finite field of \(p\) elements. A nontrivial upper bound on \(S_\chi(a, I, G) \) can be derived from the Burgess bound if \(H \ge p^{1/4+\varepsilon}\) and from some standard elementary arguments if \(T \ge p^{1/2+\varepsilon}\), where \(\varepsilon>0\) is arbitrary. We obtain a nontrivial estimate in a wider range of parameters \(H\) and \(T\). We also estimate double sums \( T_\chi(a, G) = \sum_{\lambda, \mu \in G} \chi(a + \lambda + \mu), \qquad 1\le a < p-1, \) and give an application to primitive roots modulo \(p\) with \(3\) non-zero binary digits. DOI:- 10.1017/S0004972714000227

Published

2014-09-20

Issue

Vol. 90 No. 3 (2014)

Section

Articles
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