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

TL;DR

This paper derives a detailed asymptotic expansion for the Riesz energy of equally spaced points on the unit circle as the number of points grows large, revealing connections to the Riemann zeta function.

Contribution

It provides the first complete asymptotic expansion of the Riesz energy for large N, including all powers of N and analytic continuation for complex s.

Findings

Asymptotic expansion expressed in powers of N

Optimal energy configurations for s ≥ -2

Connection to the Riemann zeta function

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 $1/r^{s}$, $s\neq0$, 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.