This is probably too hard for math.stackexchange, so I migrated it here.
For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k | $. I want to show that $H$ is globally maximized by some picket fence configuration, that is, $\lambda_j = e^{2 \pi i (\alpha + j/n)}$, for some $\alpha \in \mathbb{R}$.
Let $\lambda_j = e^{i \theta_j}$, with $0 \le \theta_1 \le \ldots \le \theta_n < 2 \pi$. Using the identity $| e^{i \theta_1} - e^{i \theta_2}| = | 2 \sin \frac{\theta_1 - \theta_2}{2}|$ and the fact that $0 \le \theta_k - \theta_j < 2 \pi$, I reduce the problem to maximizing over $\theta_j$ satisfying the above constraints, the following function: $$ \sum_{j < k} \sin \frac{\theta_j - \theta_k}{2}.$$ Assuming $\theta_j$'s are distinct for now, differentiating gives me $$ \sum_{\ell < k} \cos \frac{\theta_\ell - \theta_k}{2} - \sum_{j < \ell} \cos \frac{\theta_j - \theta_\ell}{2} = 0, \qquad \forall \ell=1,\ldots, n.$$
But how do I finish?