We discuss here the topic of angular lattice sums, i.e., those which depend on the angle or angles between the vector linking the origin to lattice points and the coordinate axes. This topic is an old one, dating back to an 1892 paper by Lord Rayleigh [26], but is curiously disconnected from the main thread of investigations into lattice sums, as surveyed by Glasser and Zucker [9]. We begin with a brief account of the history of the sums and go on to give an account of some of their more recently discovered properties. We use the latter topic to discuss how properties and formulae for lattice sums may be discovered with the aid of modern symbolic algebra packages such as Mathematica or Maple. Chief among the properties of the angular lattice sums in two dimensions that we describe is their relationship to the Riemann zeta function; selected sums obey the celebrated Riemann hypothesis.

Optical properties of coloured glass and lattice sums

The technology of colouring glass by adding to the melt appropriate metals is an old one, dating back to the time of the ancient Greeks and Romans and predating the modern field known as plasmonics. Michael Faraday proposed a model of atoms as being like tiny metal particles and so opened up the question as to what could be inferred about the properties of atoms from the interaction of light with various solids.