Does there exist a formula for product of the primitive $ n $th roots of unity. I know there is a formula for the sum of the primitive $ n $th roots of unity which is the Mobius function of $ n $. See: The M&#246;bius function is the sum of the primitive $n$th roots of unity.

I am curious about the existence of a formula for the product of the primitive $ n $th roots of unity.
If $g$ is a primitive $n$-th root of unity, then so is $g^{-1}$.

Unless $n\le2$, we have $g \ne g^{-1}$. When $n=2$, there is only one primitive $n$-th root of unity: $-1$.

Therefore, the product of all primitive $n$-th roots of unity is $1$ when $n\ne 2$ and $-1$ when $n=2$.

This is quite similar to Wilson's theorem for abelian groups:


  The product of all elements in a finite abelian group is $1$, except when there is exactly one element of order $2$, in which case the product is this element.
That's much easier than the sum!  The "primitive nth roots of unity" are $e^{\frac{ik\pi}{n}}$ for k from 1 to n.  Their product is $e^{(1+ 2+ ...+ n)(ik\pi)}= e^{\frac{n(n+1)ik\pi}{2}}$.