The hexagonal lattice achieves the extrema for ratios of theta and Epstein zeta functions.
B´ etermin, Local variational study of 2d lattice energies and application to Lennard-Jones type interactions, Nonlinearity31(9) (2018), 3973-4005
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On ratios of theta functions
The hexagonal lattice achieves the extrema for ratios of theta and Epstein zeta functions.