On the mean value of the mixed exponential sums with Dirichlet characters and general gauss sum

摘要

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers m, n, k, q, with k ⩾ 1 and q ⩾ 3, and Dirichlet characters χ, χ̄ modulo q we define a mixed exponential sum $$C(m,n;k;chi ;overline chi ;q) = sumlimits_{a = 1}^q {chi (a){G_k}(a,overline chi )eleft( {{{m{a^k} + noverline {{a^k}} } over q}} right)} ,$$ , with Dirichlet character χ and general Gauss sum G k (a, χ̄) as coefficient, where Σ′ denotes the summation over all a such that (a, q) = 1, aā ≡ 1 mod q and e(y) = e2πiy . We mean value of $$sumlimits_m {sumlimits_chi {sumlimits_{overline chi } {C{{left| {m,n;k;chi ;overline chi ;q} right|}^4}} } } ,$$ , and give an exact computational formula for it. Keywords mixed exponential sum mean value Dirichlet character general Gauss sum computational formula MSC 2010 11L03 11L05 The research has been supported by the National Natural Science Foundation of China under Grant No. 10901128.