nt.number theory - A product of gamma values over the numbers coprime to n.

Denote

$$f(n)=prod_{k=1}^{n-1}Gamma left(frac{k}{n}right)$$ and $$F(n)=prod_{1le kle n-1, kperp n}Gamma left(frac{k}{n}right)$$
We have $f(n)=prod_{d|n}F(n)$ and therefore by Mobius inversion $F(n)=prod_{d|n}f(d)^{mu(n/d)}$

By the multiplication theorem we have $f(n)=frac{1}{sqrt{n}}(2pi)^{frac{n-1}{2}}$, so if $n$ is not a prime power

$$F(n)=prod_{d|n}left(frac{1}{sqrt{d}}(2pi)^{frac{d-1}{2}}right)^{mu(n/d)}=(2pi)^{frac{1}{2}varphi (n)}$$
The formula $F(n)=sqrt{varphi(n)+1}f(varphi(n)+1)$ follows.

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