Lattice Sums Then and Now

Abstract

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques

“…[8,13]. We use here a different, new approach, based on an idea proposed recently by Barnett and Greengard [2] for the Helmholtz two-dimensional case.…”

The Exact Evaluation of Some New Lattice Sums

“…A further oddity was that not a single reference was given. Nevertheless, it should be pointed out that Lorenz, in his paper, provided results similar to (9) . Thus,…”

“…To yield the equivalent lattice sum, subtract 1 from both sides of (27), replace q by e −t Mellin transform both sides of (27), and one recovers Lorenz's result (10). Nazimov indeed produces the equivalent of Lorenz's results (10)- (13) and several more two-dimensional results, so although they were new when given by Nazimov, now they have not added anything to our knowledge, since in [9] all possible solutions of ∑ ' (am 2 + bmn + cn 2 ) −s capable of being expressed as sums of products of pairs of Dirichlet series were found and given. However, Nazimov goes on to consider many 4, 6, and even a 12 term quadratic forms, all of which can be transformed into lattice sums not previously found, and these will now be given.…”

“…Along with properties of Dirichlet L-series and many other aspects of lattice sums, these are given with numerous references in the book Lattice sums then and now [9].…”

Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series

“…We first consider a class of general real character L-series (see [12,14] and [26, §27.8]). For d ≥ 3 we employ the multiplicative characters χ ±d (n) := ±d n in terms of the generalized Legendre-Jacobi symbol, and for later use we set χ 1 (n) := 1, χ −2 (n) := (−1) n−1 , so that L 1 := ζ, while L −2 := η, the alternating zeta function.…”