An explicit formula for the fourth moment of certain exponential sums
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 \(m\).
Gaussian sums, Kloosterman sums, fourth power mean
scientific article; zbMATH DE number 5931411
scientific article