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$\boxed{\text{Conjecture}:}$ Let the sum of primitive roots of the equation $z^n-1=0$ be $C$. Let $P$ and $L$ be such that $n= P \times L $ and $\gcd (P,L)=1$ . Let the sum of primitive roots of $z^P-1=0$ be $k$ and sum of primitive roots of $z^L-1=0$ be $m.$ Now $k , m \in [0,1,-1]$. Then , $$C= k \times m.$$

Note : I found this pattern in complex numbers today. I am not aware if this an existing theorem or not. I dont seem to find this anywhere.

Aleph
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    This is multiplicativity of the Möbius function. Here ( https://math.stackexchange.com/questions/1892410/the-m%C3%B6bius-function-is-the-sum-of-the-primitive-nth-roots-of-unity ) is a proof that $\mu(n)=\begin{cases} 0&\text{if }n\text{ is not squarefree}\ (-1)^{\text{# prime divisors}}&\text{if }n\text{ squarefree}\end{cases}$ is actually that sum. It's then clear that this $\mu$ is multiplicative. – Sassatelli Giulio Oct 15 '22 at 09:38
  • Also here – Jean Marie Oct 15 '22 at 12:20

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