Bounds of Gauss Sums in Finite Fields

We consider Gauss sums of the form Gn(a) = ∑ x∈Fpm χ(x) with a nontrivial additive character χ 6= χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n−1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of HeathBrown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is nontrivial for the values of n of order up to pm/2+1/6. We also show that for almost all primes one can obtain a bound which is nontrivial for the values of n of order up to pm/2+1/2.