An explicit formula for the fourth moment of certain exponential sums (Q555505)

Consider the generalised Kloosterman sum \[ S(m,n,\chi,\chi',q) = \mathop{{\sum}'}_{a=1}^q \chi(a)G(a,\chi') e\left({ma^k+na\over q}\right) \] for integers \(m, n, q, k\) with \(q, k\geq 1\) and Dirichlet characters \(\chi,\chi'\bmod q\) and the Gaussian sum \[ G(a,\chi')=\mathop{{\sum}'}_{u=1}^q \chi'(u)e\left({ua\over q}\right). \] The main result is an exact formula for the fourth moment \(M_k(q)\) of \(S(m,n,\chi,\chi',q)\) when averaged over the parameters \(m, \chi, \chi'\). The authors note that the sums are closely related to certain mixed exponential sums considered by \textit{H. N. Liu} [Proc. Am. Math. Soc. 136, 1193--1203 (2008; Zbl 1145.11063)] because \(S(m,n,\chi,\chi',p^\alpha)= G(1,\chi')C(m,n,\chi\overline{\chi' }, k,p^\alpha)\) where \(C\) is Liu's sum \[ C(m,n,\chi,k,q)=\mathop{{\sum}'}_{a=1}^q \chi(a) e\left({ma^k+na\over q}\right) \] and the theorem also follows from Liu's result giving the fourth moment of \(C(m, n, \chi, k,q)\) averaged over \(\chi\) and \(m\).