Asymptotic expansion of the minimal Riesz 2-energy on the unit sphere

Prove the conjectured asymptotic expansion for the minimal Riesz 2-energy on the unit sphere S^2: E_2(S^2; N) = (1/4) N^2 log N + C_{2,2} N^2 + o(N^2) as N → ∞, where the constant C_{2,2} equals (1/4)(γ − log(2√3 π)) + (√3/(4π))(γ_1(2/3) − γ_1(1/3)), with γ the Euler–Mascheroni constant and γ_1(α) the first generalized Stieltjes constant in the Laurent expansion of the Hurwitz zeta function ζ(s, α) about s = 1.

Background

The paper studies discrete minimal logarithmic and Riesz energies on the unit sphere. For d = 2, sharp leading-order terms are known for logarithmic energy, and lower bounds have been recently improved using inner product expansions and connections to Riesz energies.

A key step toward better bounds is a conjectured precise asymptotic for the minimal Riesz 2-energy on S^2. Establishing this expansion and the explicit constant C_{2,2} would clarify the relationship between coefficients in different energy asymptotics and underpin improved bounds for logarithmic energy via summation-by-parts connections.