Abstract
We estimate double sums Sχ(a, I, G) = ∑x∈I∑λ∈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 language | English |
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Pages (from-to) | 376-390 |
Number of pages | 15 |
Journal | Bulletin of the Australian Mathematical Society |
Volume | 90 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2 Dec 2014 |
Externally published | Yes |
Keywords
- Character sums
- Intervals
- Multiplicative subgroups of finite fields