Asymptotics for minimal discrete energy on the sphere

We investigate the energy of arrangements of N N points on the surface of the unit sphere S d S^d in R d + 1 Rᵈ⁺¹ that interact through a power law potential V = 1 / r s , V = 1/r^s , where s > 0 s > 0 and r r is Euclidean distance. With E d ( s , N ) E_d(s,N) denoting the minimal energy for such N N -point arrangements we obtain bounds (valid for all N N ) for E d ( s , N ) E_d(s,N) in the cases when 0 > s > d 0 > s > d and 2 ≤ d > s 2 ≤ d > s . For s = d