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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
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