On certain character sums

Abstract

Let q ≥ 2 be an integer, _X 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

T_k(x, A, B,H; q) =       ∑           a^k _X (a),

                             a∈h(A,B,H)

where h(A,B,H) = {a ∈ Z| (a,q) = 1, 1 ≤ a ≤ A ≤ b ≤ B, ab ≡ (mod, q), |a-b| ≤ H}

Keywords: Character sums, Kloosterman sums, trigonometric sums