  The paper Brauchart, Hardin and Saff [Bull. Lond. Math. Soc. 41(4) (2009)]
gives the complete asymptotic expansions of the Riesz $s$-energy of the $N$th
roots of unity which form a universally optimal distribution of points on the
unit circle in the sense of Cohn and Kumar [J. Amer. Math. Soc. 20 (2007)].
Here, exact formulas (valid for all $N \geq 2$) are obtained for the case when
$s$ is an even integer. In the case of the singular Riesz $s$-potential
$1/r^s$, $r$ the Euclidean distance between two points, a continuous modified
energy approximation of the Riesz energy is used. Stirling numbers of the first
kind, Eulerian numbers and special values of partial Bell polynomials play a
central role. Several identities between these quantities are shown.
