Asymptotic properties of generalized Chaba and Pathria lattice sums

Abstract

For an arbitrary Bravais τ lattice in an m-dimensional Euclidean space amd for 0<a<∞, we present an extension of the Chaba and Pathria method of evaluating the lattice sums Σ′ττ−2k exp(−aτ2) from integral k values to all positive k values. We use the extension to study the asymptotic properties of these sums as the parameter a approaches zero. The leading term is given by πm/2 [Γ (k)(m/2−k) am/2−k]−1 for 0<2k<m, by πm/2Γ (k)−1 ln(1/a) for 2k=m, and by Σ′ττ−2k for 2k≳m. Thus 2k=m gives a transition point from structure independence to structure dependence.