Introduction

It has been more than 100 years since Appell [2] introduced lattice sums into physics, yet the article on which this chapter is based is apparently the first devoted entirely to the subject. We are, of course, aware that parts of other reviews (such as those by Born and Göppert-Mayer [21], Waddington [135], Tosi [133], and Sherman [125]) have dealt with Coulomb sums in ionic crystals, as a casual reading of this chapter will demonstrate. In the perusal of 100 years of literature, we will inevitably have missed or ignored relevant papers and their authors are urged to communicate with us directly.

The organization of this review is as follows. In Section 1.2 we present a historical survey, picking out and describing in detail some of the more important methods for calculation. Section 1.3 deals with the representation of lattice sums as Mellin-transformed products of theta functions. In Section 1.4 we discuss the evaluation of two-dimensional lattice sums by number-theoretic means, and in Section 1.5 we examine a promising new application of contour integration.