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 $N\to \infty$\mathrm{.}For$s\ge -2$,suchpointsformoptimalenergy$N$-point configurations with respect to the Riesz potential $1\mathrm{/}{r}^s$, $s\mathrm{\ne }0$, 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.