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Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist?

More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: $G=\sum_{i=0}^{p-1}\zeta^{i^2}$. We know that $|G|=\sqrt p$. I am looking for $a_0,\ldots,a_{p-1}\in N_0$: $∑a_i=p;|\sum_{i=0}^{p−1}a_i\zeta^i|=\sqrt p$

I would be surprised if this problem had not previously been studied. Has anybody seen it before?

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    $\begingroup$ I think you should be more specific about "this property": what exactly kind of object are you looking for? Also, there are lots of things that are commonly referred to as "Gauss sums". $\endgroup$
    – Alex Degtyarev
    Commented Nov 18, 2014 at 10:28
  • $\begingroup$ If you want a generalization of the Gauss sum $\sum_{a \in F_p} \left( \frac{a}{p} \right) \zeta^{a}$, then wouldn't the right condition be $\sum_{i} a_i = 0?$ After all, the Legendre symbols sum to $0$. $\endgroup$
    – Dylan Yott
    Commented Nov 19, 2014 at 6:36

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A paper by Cavior does just that. It also follows from the method in Elkies' answer to a previous MO-question.

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