Explicit formulas for the fourth moment of mixed exponential sums

We use elementary methods to obtain an explicit formula for the fourth power mean∑m=1q∑χmodq|∑a=1q′χ(a)e(mak+naq)|4, where e(y)=e2πiy, χ is a Dirichlet character modulo q and ∑′a=1q denotes the summation over all a with (a,q)=1. This extends a result of Z.F. Xu et al. by avoiding the restriction (k,q)=1.