Mean value of some exponential sums and applications to Kloosterman sums (Q1036172)

Let \(q,m,n,k\) be integers with \(q \geq 3\), \(k\geq 1\). Define \[ S(m,n,k;q) = \sideset\and {'}\to\sum_{a =1}^q e((ma + na^k)/q), \] where \(e(x) = e^{2\pi ix}\) and \(\sideset\and {'}\to\sum\) denotes summation under the condition \((a, q) =1\). It is easily seen that \[ \sideset\and {'}\to\sum_{ n=1}^q |S(m,n,k;q)|^r \] is multiplicative in \(q\) for any \(r\). The author evaluated in this paper the fourth power moment \[ \sideset\and {'}\to\sum_{ n=1}^q |S(m,n,k;q)|^4 \] for \(k =1, 2\) and \(k \equiv -1 \pmod {\varphi (q)}\). In the last case, \(S(m,n,k;q) = K(n,m;q)\), the classical Kloosterman sum. Hence a fourth power moment formula for \(K(n,m;q)\) follows from the main theorem in this paper. Also two other immediate corollaries for fourth power moments of hyper-Kloosterman sums are derived. The bulk of the paper concerns the enumeration of residue classes modulo prime powers.