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A note on the Dirichlet characters of polynomials. (English) Zbl 1076.11048
This paper is a continuation of a paper of the first author together with Y. Yi [Bull. Lond. Math. Soc. 34, 469–473 (2002; Zbl 1038.11052)]. The main result is a generalized identity of the form $\sum_{n=1}^{q}\chi\bigl(f(n)\bigr)= \varepsilon(\chi,f)\cdot q^{1/2}$ where $$q$$ is a perfect square, $$\chi$$ a primitive Dirichlet character modulo $$q$$, $$f$$ a certain polynomial with integer coefficients and $$|\varepsilon(\chi,f)|=1$$ is explicitly given. As a corollary they got for $$q$$ an odd square number and $$m$$ any natural number $$m$$ with $$(q,m)=1$$ the following nice identity $\sum_{n=1}^{q}\chi\bigl(n^m(1-n)^m\bigr)= \overline{\chi}(4^m)\cdot q^{1/2}.$ For general moduli $$q$$, whether there exists a similar formula, is an open problem.

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