The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere

TL;DR

This paper reviews known results and proposes conjectures for the next-order term in the asymptotic expansion of optimal Riesz and logarithmic energies of points on spheres, based on analytic continuation assumptions.

Contribution

It introduces new conjectures for the next-order asymptotic terms in Riesz and logarithmic energy problems on spheres, extending existing results.

Findings

Survey of known asymptotic results

Proposed conjectures for next-order terms

Analytic continuation assumptions underpinning conjectures

Abstract

We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz $s$-energy of $N$ points on the unit sphere in $\mathbb{R}^{d+1}$, $d\geq 1$. The conjectures are based on analytic continuation assumptions (with respect to $s$) for the coefficients in the asymptotic expansion (as $N\to \infty$) of the optimal $s$-energy.