On the fourth power mean of the general Kloosterman sums (Q5907088)

Let \(p\) be an odd prime. For any fixed integer \(n\) coprime with \(p\) the author gives an exact calculating formula for the expression \[ \sum_{m=1}^p ~\left| ~\sum_{a=1}^{p-1} \, \chi(a) \, e^{2 \pi i \frac{ma+n \bar{a}}{p}} ~\right| ^4 ~~. \] Here, \(\chi\) denotes a Dirichlet character mod \(p\) and \(\bar{a}\) satisfies \(a \bar{a} \equiv 1\) mod \(p\) .