The authors show that ∂²/∂x∂y of the theta and Epstein zeta functions is strictly positive, and ∂³/∂x∂y² is strictly negative, in the relevant fundamental domain.
Minimizing Lattice Energy and Hexagonal Crystallization
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abstract
Consider the energy per particle on the lattice given by $\min_{ \Lambda }\sum_{ \mathbb{P}\in \Lambda} \left|\mathbb{P}\right|^4 e^{-\pi \alpha \left|\mathbb{P}\right|^2 }$, where $\alpha >0$ and $\Lambda$ is a two dimensional lattice. We prove that for $\alpha\geq\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\frac{\pi}{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by B\'etermin.
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Signs of high order derivatives for the theta and Epstein zeta functions and application
The authors show that ∂²/∂x∂y of the theta and Epstein zeta functions is strictly positive, and ∂³/∂x∂y² is strictly negative, in the relevant fundamental domain.