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Bounds for certain exponential sums. (English) Zbl 1030.11040
The exponential sums in question are of the type $$S=\sum_{x \pmod{p^m}} e((ax^n+bx)/p^m)$$, where $$p^m$$ is a prime power and $$n\geq 2$$; also, twists with Dirichlet characters are considered. The main result is the following estimate. Let $$n \geq 2$$, $$m \geq 2$$, $$h=\text{ord}_p(n-1)$$, $$\beta = \text{ord}_p(n)$$, $$\tau = \text{ord}_p(a)$$, and suppose that $$\tau \leq m-2$$. Then for $$p>2$$ $|S|\leq (n-1,p-1)p^{(1/2)(\min (1,\beta)+ \min (h,m-2-\tau)+m)}(b,p^m)^{1/2},$ and for $$p=2$$ $|S|\leq 2 p^{(1/2)(\min (h, m-2-\tau)+m)}(b,p^m)^{1/2}.$ Kloosterman sums with the denominator $$p^m$$ represent a special case of the sums $$S$$, and a classical estimate of Estermann for such sums follows as a corollary from the above estimate for $$p > 2$$. In addition, estimates for hyper-Kloosterman sums are given.

##### MSC:
 11L07 Estimates on exponential sums 11L05 Gauss and Kloosterman sums; generalizations
##### Keywords:
exponential sums; Kloosterman sums
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