On certain character sums

Abstract

Let q ≥ 2 be an integer, χ be a non-principal character mod q, A = A(q) ≤ q, B = B(q) ≤ q, and H = H(q) ≤ q. In this paper, we shall use the estimates for Kloosterman sums and the properties of trigonometric sums to give some sharp estimates for character sums of the form where ħ(A, B, H) = {a ∈ ℤ| (a, q) = 1, 1 ≤ a ≤ A, 1 ≤ b ≤ B, ab ≡ 1 (mod q), |a − b| ≤ H}.