4.2.1 Example: \(N = 3\), \(j = 2\)
For \(N = 3\) and \(j = 2\), the power sum is
\begin{equation} S_2(\theta ) = \sum _{k=0}^2 \cos ^2\left(\theta + \frac{2\pi k}{3}\right). \end{equation}
3
Using \(\cos ^2\theta = (1 + \cos (2\theta ))/2\):
\begin{equation} S_2(\theta ) = \frac{3}{2} + \frac{1}{2} \sum _{k=0}^2 \cos \left(2\theta + \frac{4\pi k}{3}\right). \end{equation}
4
The sum vanishes by Lemma 3.1.2 (with \(m = 2 {\lt} N = 3\)), giving \(S_2(\theta ) = 3/2\) for all \(\theta \).