Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies

Abstract

We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H. Montgomery, Minimal theta functions. Glasgow Mathematical Journal, 30 (1988), 75–85]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery’s result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating charged lattices and lattices shifted by the center of their unit cell.