The Brauchart-Hardin-Saff conjecture on the linear term of optimal logarithmic energy on the 2-sphere

Corollary 1.3: $E_{\log}(n)=(\tfrac12-\log2)n^2-\tfrac n2\log n+C_{\mathrm{BHS}}n+o(n)$ with the Brauchart-Hardin-Saff constant, obtained from the triangular minimality of the planar Coulomb renormalized energy and the Betermin-Sandier equivalence. It is the logarithmic, $d=2$ case only; the Riesz-energy cases of the Brauchart-Hardin-Saff conjectures and higher spheres are not addressed, and no explicit point configurations are produced.

Corollary 1.3: $E_{\log}(n)=(\tfrac12-\log2)n^2-\tfrac n2\log n+C_{\mathrm{BHS}}n+o(n)$ with the Brauchart-Hardin-Saff constant, obtained from the triangular minimality of the planar Coulomb renormalized energy and the Betermin-Sandier equivalence. It is the logarithmic, $d=2$ case only; the Riesz-energy cases of the Brauchart-Hardin-Saff conjectures and higher spheres are not addressed, and no explicit point configurations are produced.