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Is there any unsolved problem related Gauss sum or more generally some kind of a sum of roots of unity?

Also I would like to know if there is an unsolved problem that can be proved if some (unproved) property of a sum of roots of unity is shown.

It is a quite vague question but I'd like to know connections between many areas.

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How small in modulus can a non-zero sum of 5 $n$th roots of unity be? There's a simple argument that shows the modulus is greater than (essentially) $5^{-n/2}$, but computations suggest it never gets anywhere near that small, and maybe never gets smaller than about $n^{-3}$. There are infinitely many $n$ for which the modulus can go as low as $Cn^{-2}$ for some constant $C\gt0$. I wrote a short paper about this (and related problems) in the Monthly about 30 years ago.

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Yes. The determination of Gauss sums with characters of prime order at least 5 is an unsolved problem.

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