We prove that for α ≥ 1, among 2d unit density lattices, is achieved at hexagonal lattice for and does not exist for . Here the hexagonal lattice with unit density can be expressed by . This leads to two applications as follows. (1) Assume that α ≥ 1. Then, among 2d unit density lattices, is achieved at hexagonal lattice. (2) Assume that β > α ≥ 1. Then is achieved at (corresponding to hexagonal lattice) for and does not exist for . Here θ(α; z) is the two-dimensional Theta function.
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