The hexagonal lattice achieves the extrema for ratios of theta and Epstein zeta functions.
B´ etermin, Two-dimensional theta functions and crystallization among Bravais lattices,SIAM Journal on Mathematical Analysis, 48(5) (2016), 3236-269
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On ratios of theta functions
The hexagonal lattice achieves the extrema for ratios of theta and Epstein zeta functions.