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Bounds for incomplete hyper-Kloosterman sums. (English) Zbl 0923.11118

For a prime \(p \geq 3\), the “complete” hyper-Kloosterman sum is \[ Kl _m(p)=\sum_{d_1=1}^{p-1}\cdots \sum_{d_m=1}^{p-1}e\left (\frac{d_1+\dotsb +d_m+\overline{d_1\dotsb d_m}}{p}\right), \] where the bar indicates multiplicative inverse \(\pmod{p}\). In the corresponding “incomplete” sum \(Kl_m(p, x_1, \dotsc ,x_m)\), each variable \(d_i\) runs, respectively, over the interval \([1, x_i]\). The following bound is obtained for the latter sum: \[ Kl _m(p, x_1, \dotsc ,x_m) \ll x_1\dotsb x_m/p + p^{1/2}(x_1\dotsb x_m)^{1-1/r}(p^{1/4(r-1)}\log ^2p)^m \] for any fixed integer \(r\geq 2\). The proof is based on estimates of D. A. Burgess [Bull. Lond. Math. Soc. 20, 589-592 (1988; Zbl 0667.10024)] for incomplete Gauss sums.
Reviewer: M.Jutila (Turku)

MSC:

11L05 Gauss and Kloosterman sums; generalizations

Citations:

Zbl 0667.10024
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References:

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