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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?

lhf
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    Related: http://math.stackexchange.com/questions/25452/prove-sum-of-primitive-roots-congruent-to-mup-1-pmodp – JavaMan Jan 12 '12 at 16:08
  • @JavaMan, thanks, I've seen this and I have added my question there as a comment but I though an actual question would be more visible. Moreover, that question you mentioned does not touch my question. – lhf Jan 12 '12 at 16:11
  • After I included the link, I saw that you commented there. I wasn't sure whether to keep the link here or not, but I agree that asking this as a new question will attract more attention. – JavaMan Jan 12 '12 at 16:47
  • Is there a reason to think that it is particularly interesting or useful? I'm not saying it's not, but it looks like a homework problem relating a cute fact you get by looking at coefficients of minimal polynomials... – Cam McLeman Jan 12 '12 at 17:11
  • I think its main use is as a neat exercise you can give to a number theory class to see whether they figure out how to apply the concepts you've taught. – Gerry Myerson Jan 15 '12 at 11:51
  • @Gerry, Wikipedia says this result was found by Gauss. He probably found it interesting. Perhaps I should look in Disquisitiones Arithmeticae (art. 81)... – lhf Jan 24 '12 at 15:28
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    It can never be a bad idea to look at the Disquisitiones. – Gerry Myerson Jan 25 '12 at 01:35
  • A generalization of Gauss's results is given in G. A. Miller, On the Sum of the Numbers Which Belong to a Fixed Exponent as Regards a Given Modulus, The American Mathematical Monthly, Vol. 19, No. 3 (Mar., 1912), pp. 41-46. At the end of that paper, there are a few useful applications. – lhf Jan 25 '12 at 10:01
  • But Gauß had successfully applied the sum of primitive roots to prove some important facts of the theory of quadratic forms, further leading to the prototype of the theory of characters, right? I am not sure if this is correct, but it appears so... – awllower Feb 08 '12 at 09:15
  • @awllower, ah, if you can provide a reference, please add this observation as an answer. Thanks. – lhf Feb 08 '12 at 09:46
  • @Ihf: Surely, when I can again get my hands on that book, an answer should be within the options. But the veracity of this naïve idea still needs to be vouched. – awllower Feb 08 '12 at 10:33
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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. – Frank Hubeny Feb 25 '16 at 16:01

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I'll give one place where I used this result in representation theory.

The vector space Sym$^d(V)$ where $V$ is the natural degree n represntation of $S_n$ is regarded as a permutation representation for the dihedral group $D_n$. Being reducible, we computed the dimension of the isotypical components of this representation. This dimensions is a sum involving character values which are of the form $\cos 2\pi k/n$ with $k$ coprime to $n$.

We noted that if the sum of primitive $n$th roots of unity is $\mu(n)$ then the sum of their real parts is also $\mu (n)$. The real parts being these cosine values lead to a nice dimension formula.

This result is available in the link: https://www.ias.ac.in/article/fulltext/pmsc/130/0016

P Vanchinathan
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