Mean value of mixed exponential sums

Abstract

For integers q, m, n, k with q,k ≥ 1, and Dirichlet character Χ mod q, we define a mixed exponential sum C(m, n, k,Χ,q):= Σ q a=1 Χ(α) (ma k + na) q, where e(y) = e 2πij , and Σ ' α denotes the summation over all a with (a, q) = 1. The main purpose of this paper is to study the mean value of q Σ Σ'|C(m,nknΧ;q| 4 , Χmod q m=1 and to give a related identity on the mean value of the general Kloosterman K(m, n, Χ; q) Σ q α=1 Χ(α)e (ma+na / q), where aa ≡ 1 mod q.