  We study the Hamiltonian of a two-dimensional log-gas with a confining
potential $V$ satisfying the weak growth assumption -- $V$ is of the same order
than $2\log|x|$ near infinity -- considered by Hardy and Kuijlaars [J. Approx.
Theory, 170(0) : 44-58, 2013]. We prove an asymptotic expansion, as the number
$n$ of points goes to infinity, for the minimum of this Hamiltonian using the
Gamma-Convergence method of Sandier and Serfaty [Ann. Proba., to appear, 2015].
We show that the asymptotic expansion as $n\to +\infty$ of the minimal
logarithmic energy of $n$ points on the unit sphere in $\mathbb{R}^3$ has a
term of order $n$ thus proving a long standing conjecture of Rakhmanov, Saff
and Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the
equivalence between the conjecture of Brauchart, Hardin and Saff [Contemp.
Math., 578:31-61,2012] about the value of this term and the conjecture of
Sandier and Serfaty [Comm. Math. Phys., 313(3):635-743, 2012] about the
minimality of the triangular lattice for a "renormalized energy" $W$ among
configurations of fixed asymptotic density.
