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Dirichlet character sums. (English) Zbl 0931.11029
Let \(\chi\) be a Dirichlet character of conductor \(p^n\) with \(p \in {\mathbb{P}}\), \(n \in {\mathbb{N}}\), and let \(f(x) = a_0 + a_1 x + \cdots + a_k x^k\) be an integral polynomial such that \(k>3\) and \((p^n,a_1,\ldots,a_k)= p^m\). Using a special iteration the author proves some general character sum estimates of type \[ p^{-(n-m)(1-1/k)} \Big |\sum_{0 \leq x < p^{n-m}} \chi(f(x)) \Big |\leq \begin{cases} 1 &\text{if } p \geq (k-1)^{2k/(k-2)}\\ k & \text{ otherwise}\end{cases}. \]
Reviewer: J.Hinz (Marburg)

MSC:
11L15 Weyl sums
11L40 Estimates on character sums
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