of Saff and Kuijlaars [@PointsSphere] (see also [@BrauchGrab]). For instance Smale’s $7^{th}$ problem [@231301] is to find, for any $n\geq 2$, a universal constant $c\in{\mathbb R}$ and a nearly optimal configuration $(y_1,...,y_n)\in (\S^2)^n$ such that, letting $\mathcal{E}_{\log}(n)$ denote the minimum of $\textnormal{E}_{\log}$ on $(\S^2)^n$, $$\textnormal{E}_{\log}(y_1,...,y_n)-\mathcal{E}_{\log}(n)\leq c\log n.$$ Identifying the term of order $n$ in the expansion of $\mathcal{E}_{\log}(n)$ can be seen as a modest step towards a better understanding of this problem.\
It was known (lower bound by Wagner [@57451281] and upper bound by Kuijlaars and Saff [@403978]), that $$\left(\frac{1}{2}-\log 2\right)n^2-\frac{1}{2}n\log n+c_1 n\leq \mathcal{E}_{\log}(n)\leq \left(\frac{1}{2}-\log 2\right)n^2-\frac{1}{2}n\log n+c_2 n$$ for some fixed constant $c_1$ and $c_2$. Thus one can naturally ask for the existence of the limit $$\lim_{n\to +\infty}\frac{1}{n}\left[\mathcal{E}_{\log}(n)-\left(\frac{1}{2}-\log2\right)n^2+\frac{n}{2}\log n  \right].$$

**CONJECTURE 1** (Rakhmanov, Saff and Zhou, [@2400418]): There exists a constant $C$ not depending on $n$ such that $$\mathcal{E}_{\log}(n)=\left(\frac{1}{2}-\log2  \right)n^2-\frac{n}{2}\log n+ Cn+o(n) \quad \text{ as } n\to +\infty.$$

**CONJECTURE 2** (Brauchart, Hardin and Saff, [@Brauchart]): The constant $C$ in Conjecture 1 is equal to $C_{BHS}$, where $$\label{CBHSdef}
C_{BHS}:=2\log2 +\frac{1}{2}\log\frac{2}{3}+3\log\frac{\sqrt{\pi}}{\Gamma(1/3)}.$$

As we will see, our results imply that the last conjecture is equivalent to one concerning the global optimizer of the renormalized energy $W$.

**CONJECTURE 3** (Sandier and Serfaty, [@Sandier_Serfaty], or 