Discrete energy asymptotics on a Riemannian circle

We derive the complete asymptotic expansion in terms of powers
of N for the geodesic f -energy of N equally spaced points on a rectifiable simple
closed curve Γ in Rp, p ≥ 2, as N → ∞. For f decreasing and convex, such a
point configuration minimizes the f -energy ∑
j6 =k f (d(xj , xk )), where d is the ge-
odesic distance (with respect to Γ) between points on Γ. Completely monotonic
functions, analytic kernel functions, Laurent series, and weighted kernel func-
tions f are studied. Of particular interest are the geodesic Riesz potential 1/ds
(s 6 = 0) and the geodesic logarithmic potential log(1/d). By analytic continuation
we deduce the expansion for all complex values of s.