No abstract

No abstract

Abstract. We estimate double sumswith 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 ≥ p 1/4+ε and from some standard elementary arguments if T ≥ p 1/2+ε , where ε > 0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters H and T . We also estimate double sumsand give an application to primitive roots modulo p with 3 non-zero binary digits.

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface