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On Dirichlet characters of polynomials. (English) Zbl 1038.11052
The classical result, due to Pólya and Vinogradov, is that the estimates $\sum_{n=N+1}^{N+H}\chi(n)\ll q^{1/2}\log q$ holds for all nonprincipal Dirichlet characters $$\chi$$ modulo $$q$$. The main result of the note under review is to show that for certain primitive characters $$\chi$$ modulo $$q$$ and some special polynomials $$f(x)$$ with integral coefficients an identity of the form $\left| \sum_{n=1}^{q}\chi\bigl(f(n)\bigr)\right| = q^{1/2}$ holds.

##### MSC:
 11L10 Jacobsthal and Brewer sums; other complete character sums 11L40 Estimates on character sums
##### Keywords:
Dirichlet characters
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