# Simplifying a product over roots of unity

Let $\zeta_{n}=e^{2\pi i /n}$ be the nth root of unity. Now consider the product : $$\prod_{k=1}^{n-1} (1-\zeta_{n}^{k})^{\zeta_{n}^{k}}$$ Is there a simple formula for this product as a function of $n$ !?

• Note that $z^w$ is a multiple-valued function. Should we take the branch where $z$ doesn't lie on the negative real line? – Jonathan Y. Sep 6 '13 at 22:43
• I'm afraid I don't take your meaning ! – Mohammad Al Jamal Sep 6 '13 at 22:52
• Well, how do you define $(1-\zeta_n^k)^{\zeta_n^k}$? – Jonathan Y. Sep 6 '13 at 22:56
• ah.. ok ... yes, your $z$ doesn't lie on the negative real line . – Mohammad Al Jamal Sep 6 '13 at 22:59