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A note on the power mean of the general Kloosterman sums. (Chinese. English summary) Zbl 1084.11047

In this paper the authors study the generalized Kloosterman sum \[ S(m,n, \chi ; q) = \sum\limits^q_{a=1} \chi (a) e((ma+n\overline a)/q) \] where \(\chi\) is a Dirichlet character mod \(q\). Recent work of W. Duke and H. Iwaniec [A tribute to Emil Grosswald: number theory and related analysis. Providence, RI: American Mathematical Society. Contemp. Math. 143, 255–258 (1993; Zbl 0792.11029)] showed that certain cubic exponential sums are indeed \(S(m,n , \chi ; q)\) for a cubic character \(\chi \pmod q\). By using an earlier result of B. J. Birch [J. Lond. Math. Soc. 43, 57–60 (1968; Zbl 0183.25503)] on mean values of cubic exponential sums, the authors prove asymptotic formulas for the mean values \[ \sum\limits^p_{m=1} | S(m,n, \chi , p)| ^{2k},\quad k = 3, 4, 5, 6, \] where \(p\) is any prime \(\equiv 1 \pmod 3\) and \(\chi \pmod p\) is any cubic Dirichlet character.

MSC:

11L05 Gauss and Kloosterman sums; generalizations
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