Timeline for Estimating the sum of Dirichlet character $\sum_{0 \leq x < q} \chi(F(x))$ where $F(x)$ is a polynomial
Current License: CC BY-SA 3.0
10 events
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| Aug 8, 2017 at 17:03 | comment | added | reuns | I'm just saying that if $p$ is prime and $\chi$ is a Dirichlet character modulo $p$ of order $k$ then $\displaystyle\frac{1}{k}\sum_{m=1}^k \chi(x)^m = 1_{x \in \mathbb{F}_p^{* \ k}}$ and $\displaystyle\# \{ (x,y)\in\mathbb{F}_p^*, y^k = F(x)\} = \sum_{m=1}^k \sum_{x\in \mathbb{F}_p^*} \chi(F(x))^m$, so it is related to the Weil's conjectures. See Siksek's link and Iwaniec & Kowalski's book. | |
| Aug 7, 2017 at 16:45 | comment | added | Johnny T. | @reuns Could you possibly elaborate more on this? I am not familiar enough with Weil's conjecture to derive the conclusion. Thank you very much. | |
| Aug 6, 2017 at 15:05 | history | edited | Johnny T. | CC BY-SA 3.0 |
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| Aug 6, 2017 at 15:04 | comment | added | Johnny T. | @Siksek Thank you very much for all the references! | |
| Aug 6, 2017 at 9:23 | comment | added | Siksek | Another great reference is chapter 12 of the textbook of Iwaniec and Kowalski. | |
| Aug 6, 2017 at 9:15 | comment | added | Siksek | ... 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 $x$. | |
| Aug 6, 2017 at 9:10 | comment | added | Siksek | A good reference for this is the following paper of Chang: arxiv.org/abs/1201.0299 | |
| Aug 6, 2017 at 6:47 | comment | added | Gerry Myerson | Well, there is the trivial bound, $q$. If $F(x)$ is a constant, you can't do better. I think your problem is underspecified. | |
| Aug 6, 2017 at 5:34 | history | edited | Johnny T. | CC BY-SA 3.0 |
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| Aug 6, 2017 at 5:27 | history | asked | Johnny T. | CC BY-SA 3.0 |