Sarnak–Strömbergsson conjecture on cubic lattice optimality

Let $\zeta_L$ and $\theta_L$ denote the Epstein zeta function and theta function of a Bravais lattice $L$, and let $D_3$ and $D_3^*$ denote the FCC and BCC lattices, respectively, normalized to fixed unit density. For $\alpha>0$ and $s>3/2$, the relevant theta and Epstein zeta energies are considered as functions of $L$. Sarnak–Strömbergsson conjecture. For any $\alpha>\pi$ and $s>3/2$, $D_3$ is the unique minimizer of the theta and Epstein zeta functions among Bravais lattices of fixed unit density; for any $\alpha<\pi$ and $s<3/2$, $D_3^*$ is the unique minimizer among such lattices. This conjecture concerns global optimality of the cubic structures and motivates the study of their local minimality; its resolution status is not specified in the source.