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On Kloosterman’s sum. (English) Zbl 0157.09001

The author shows that if \(\chi\) is any non-quadratic character modulo the odd prime \(p\) then the generalised Kloosterman sum satisfies the identity \[ \left|\sum_{n=1}^{p-1} \chi(m)e^{2\pi i(am-m^{-1})/p}\right| = \left|\sum_{n=0}^{p-1} e^{4\pi in}\chi(m) \bar\chi(n^2+a)\left(\frac{n^2+a}{p}\right)\right| \tag{1} \] and thus he estimates the Kloosterman sum by applying the well-known estimate \(2\sqrt{p}\) (due to A. Weil) to the right-hand side. (1) has been also proved by H. Davenport in [J. Reine Angew. Math. 169, 158–176 (1933; Zbl 0006.29501)].

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11L07 Estimates on exponential sums

Keywords:

Kloosterman sum

Citations:

Zbl 0006.29501
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