A note on mean value of mixed exponential sums

Using elementary methods, we obtain an explicit formula for the fourth power mean∑m=1q′∑χmodq|∑a=1q′χ(a)e(mak+naq)|4 for arbitrary positive integer k  , where e(y)=e2πiye(y)=e2πiy, χ is a Dirichlet character modulo q   and ∑a=1q′ denotes the summation over all a   such that (a,q)=1(a,q)=1. This improves H.N. Liu's result by averting the restriction (k,q)=1(k,q)=1.