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We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V =1 =r s ; where s> 0a nd ris Euclidean distance. With Ed(s;N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N )f or E d (s;N) in the cases when 0 <s< dand 2 d< s .F or s = d, we determine the precise asymptotic behavior of Ed(d; N )a s N !1 . As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy conguration, when s d 2. For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E2(s;N) that relate to the zeta function L(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the rst of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of L(s )w hen 0 <s< 2 (the divergent case).

Asymptotics for minimal discrete energy on the sphere