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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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