On the convergence problem for lattice sums

Three-dimensional lattice sums Sigma +or-r -s , where r is the distance to a +or- charged lattice point, arise in classical lattice point problems (s=0) and the Madelung problem of physics and chemistry (s=1). In the latter case the sum over an infinite lattice is purely formal and the value of the Madelung constant must be defined precisely, e.g., via related convergent sums or analytic continuation in s. Indeed, the partial sum over the sphere r(R does not converge as R becomes large. The authors verify a conjecture of J F Delord (1988) that convergence can be obtained by neutralising each sphere with an appropriate surface charge. Specifically, if L R (s) is the sum over the lattice points in the sphere r(R then, for neutrally charged lattices, they show that as R goes to infinity the difference L R (s)-R -s L R (0) approaches L(s), where L(s) is defined by analytic continuation. When s=1 the term L(1) is the Madelung constant and L R (0)/R is the Coulombic correction term.