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It is well known that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \bmod{p}$.

But I can't see why this result is interesting or useful. Can someone please enlighten me?

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    @lhf One reason why it looks interesting to me is that $\mu (n)$ can be only three values: ${-1,0,1}$. Why should the sum of the primitive roots modulo their prime end up being only one of these three numbers? I would expect the sum to be scattered about the residue classes modulo that prime. However, that is only a partial answer. You also want to know why it is useful.2016-02-25

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