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On the fourth power mean of the general Kloosterman sums. (English) Zbl 1046.11055

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 \(p\) .

MSC:

11L05 Gauss and Kloosterman sums; generalizations
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