The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$

Abstract

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 $1/r^s$, $s\ne 0$, 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.

arXiv commentAdded reference to paper by Cohn and Kumar which shows optimality of roots of unity for -2<s<-1

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