see the review by Serfaty [@Serfaty:2013fk]) : The triangular lattice is a global minimizer of $W$ among discrete subsets of ${\mathbb R}^2$ with asymptotic density one.

The expansion in the particular case $V(x)=\log(1+|x|^2)$ transported to $\S^2$ using an inverse stereographic projection and appropriate rescaling gives an expansion for $\mathcal{E}_{\log}(n)$ and thus proves Conjecture 1. The constant $C$ in Conjecture 1 can moreover be expressed in terms of the minimum of the renormalized energy $W$. The value of $W$ for the triangular lattice obviously provides an upper bound for this minimum, and by using the Chowla-Selberg formula to compute the expression given in [@Sandier_Serfaty] for this quantity, we show that this upper bound is precisely $C_{BHS}$. This bound is of course sharp if and only if Conjecture 3 is true. Thus we deduce from the following.

There exists $C\neq 0$ independent of $n$ such that, as $n\to +\infty$, $$\mathcal{E}_{\log}(n)=\left(\frac{1}{2}-\log2  \right)n^2-\frac{n}{2}\log n+Cn+o(n),\quad  C= \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\log2.$$ Moreover $\displaystyle C\leq C_{BHS}$ where $C_{BHS}$ is given in , and equality holds iff $\displaystyle \min_{\mathcal{A}_1}W$ is achieved for the triangular lattice of density one.

The plan of the paper is as follows. In Section \[RE\] we recall the definition of $W$ and some of its properties 