A variational principle for Gaussian lattice sums

Source abstract

Abstract We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices Λ ⊂ R 2 $\Lambda \subset \mathbb {R}^2$ normal upper Lamda subset of double struck upper R squared with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery’s 1988 result, which proved that the hexagonal lattice minimizes the maximum.