Timeline for Non-standard Gauss sums
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 19, 2015 at 21:47 | comment | added | Liss | Ok, thank you! I really appreciate your intuition and all the hints. This is not my field, so I was not sure if this was a feasible question. It was worth trying though :), and learning a small bit of the beautiful mathematics behind those questions.. | |
| May 19, 2015 at 21:24 | comment | added | GH from MO | @Liss: I am pretty sure your sums are not bounded away from zero, i.e. they can be arbitrary small for large $p$. (I don't have a proof though.) | |
| May 19, 2015 at 21:18 | comment | added | Liss | thank you, @GHfromMO, this is also interesting approach to know. I already have found similar bound in the work of Konyagin, S.V. and Lev, V.F. (2000) "On the distribution of exponential sums". Theorem 1 there says that a sum of roots of unity taken over a set of $n$ elements is bounded in absolute value from below by $n^{-(p-3)/4}.$ I can use this result to show that the full sum from the beginning of the question (with the prescribed $\chi(k)$) is bounded by $(\frac{p-3}{4})^{-(p-3)/4}.$ Still, as I wrote in some other comment, I was curious to know if a bound away from zero can be obtained. | |
| May 19, 2015 at 4:04 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 19, 2015 at 3:58 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 19, 2015 at 3:46 | comment | added | Alexey Ustinov | $$\sum_{x=1}^p \biggl(\frac{(x-a)(x-b)}{p}\biggr)=p\delta_p(a-b)-1.$$ | |
| May 19, 2015 at 3:41 | comment | added | Lucia | Yes indeed! I forgot about my comment! | |
| May 19, 2015 at 3:33 | comment | added | GH from MO | @Lucia: I think Elkies proves that the Kloosterman sum is not in the prime ideal $(1-\omega_p)$, which rules out zero. Anyways, I gave a direct argument based on your version of Elkies's proof, please check! (I am too sleepy now.) | |
| May 19, 2015 at 3:25 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 19, 2015 at 3:20 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 19, 2015 at 3:07 | comment | added | Lucia | Isn't Elkies's answer about ruling out the maximal size $2\sqrt{p}$ for Kloosterman sums, rather than zero? | |
| May 19, 2015 at 2:56 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 19, 2015 at 2:49 | history | answered | GH from MO | CC BY-SA 3.0 |