The Riesz energy of the $N$-th roots of unity: an asymptotic expansion   for large $N$
  We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points.   By analytic continuation we deduce the expansion for all complex values of $s$.   The Riemann zeta function plays an essential role in this asymptotic expansion. 
J. S. Brauchart, D. P. Hardin, E. B. Saff