Sum of primitive roots of unity
Posted: Tue Apr 26, 2016 8:40 pm
Let $\zeta_n$ be a primitive root of unity. Evaluate the sum:
$$\mathcal{S}= \sum_{\begin{subarray}{c} 1 \leq k \leq n \\
\gcd(k,n)=1\end{subarray}} \zeta^k_n$$
$$\mathcal{S}= \sum_{\begin{subarray}{c} 1 \leq k \leq n \\
\gcd(k,n)=1\end{subarray}} \zeta^k_n$$