The fundamental conjecture for optimal Riesz energy on spheres

Let $d\geq 2$, let $-2<s<d+2$ with $s\neq 0,d$, and let $\mathbb{S}^d$ be the unit sphere in $\mathbb{R}^{d+1}$. Write $\mathcal{E}_s(\mathbb{S}^d;N)$ for the optimal discrete Riesz $s$-energy of $N$ points on $\mathbb{S}^d$, $V_s(\mathbb{S}^d)$ for the continuous Riesz $s$-energy, and let $\mathcal{H}_d$ be normalized $d$-dimensional Hausdorff measure. Then there are a constant $C_{s,d}$ and a remainder $\mathcal{R}_s(\mathbb{S}^d;N)$ such that

$$\mathcal{E}_s(\mathbb{S}^d;N)=V_s(\mathbb{S}^d)N^2+\frac{C_{s,d}}{[\mathcal{H}_d(\mathbb{S}^d)]^{s/d}}N^{1+s/d}+\mathcal{R}_s(\mathbb{S}^d;N),$$

with

$$\frac{\mathcal{R}_s(\mathbb{S}^d;N)}{N^{1+s/d-\varepsilon}}\to 0$$

as $N\to\infty$ for some $\varepsilon>0$ possibly depending on $d$ and $s$. Fundamental conjecture. The stated asymptotic expansion holds.