Abstract
The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, the authors provide explicit expansions for the spatial potential and energy density of three-dimensional periodic structures. These formulae, involving only elementary functions, are suitable for systematic calculation of Madelung constants of arbitrary point-charge crystals. They indicate how zeta function relations may be used for dimensional reduction of certain multiple sums arising in the special cubic cases.