Fourth power mean of the general Kloosterman sum

Abstract

Let q>2 be an integer, and let x be a Dirichiet character modulo q. Let k be a positive integer. For arbitrary integers m and n, define K(m, n, k, x;q) = Σ'q a=1 X(a)e(mak + na/q ), where Σ' denotes the summation over all a with (a,q)=1,e(y)-exp(27πiy), and a is the inverse of a modulo q such that 1≤-a≤ q and aa ≡ (mod q). The fourth power mean of K(m,n, x ,q) was studied, and some identities were given.