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Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not: $$ \sum_{0 \leq x < q} \chi(F(x)). $$

I would appreciate any comments and references. Thank you very much!

PS In fact I am also interested in if a more general sum $$ \sum_{0 \leq x < q} \chi(F(x)) \overline{\chi(G(x) )} $$ can be bounded or not. Here $G$ is another polynomial with integer coefficients.

PPS For the first sum I am particularly interested in the case when $F(x)$ is not identically a constant modulo $q$.

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  • $\begingroup$ Well, there is the trivial bound, $q$. If $F(x)$ is a constant, you can't do better. I think your problem is underspecified. $\endgroup$
    – Gerry Myerson
    Commented Aug 6, 2017 at 6:47
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    $\begingroup$ A good reference for this is the following paper of Chang: arxiv.org/abs/1201.0299 $\endgroup$
    – Siksek
    Commented Aug 6, 2017 at 9:10
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    $\begingroup$ ... in particular you should look at Weil's Theorem (page 7) and Theorem $3^{\prime\prime}$ (page 16). In particular, the latter should give you a non-trivial estimate for your second sum as $\chi(F(x)) \overline{\chi(G(x))}$ can be rewritten as $\chi(F(x)/G(x))$ for all but a few values of $x$. $\endgroup$
    – Siksek
    Commented Aug 6, 2017 at 9:15
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    $\begingroup$ Another great reference is chapter 12 of the textbook of Iwaniec and Kowalski. $\endgroup$
    – Siksek
    Commented Aug 6, 2017 at 9:23
  • $\begingroup$ @Siksek Thank you very much for all the references! $\endgroup$
    – Johnny T.
    Commented Aug 6, 2017 at 15:04

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