A note on alternating series in several dimensions

In the course of a study of chemical lattice sums [1] the authors considered sums such as (1.1) ∑′ (- 1) ^n+m+k ( n² + m² + k²) ^-1/2, the summation being over all non-zero integer triples. Such "sums" occur naturally in the study of crystal potentials. For example, (1.1) is meant to measure the potential at the origin of an infinite cubic crystal with unit Coulomb charges at each integer lattice point. As such the sum is considered to represent an electrochemical constant (Madelung's constant) for sodium chloride. An excellent account of such lattice sums can be found in Glasser and Zucker's recent survey [3].