Minkowski-Hlawka Theorem

There exist lattices in $n$-D having Hypersphere Packing densities satisfying

$$\eta\geq{\zeta(n)\over 2^{n-1}},$$

where $\zeta(n)$ is the Riemann Zeta Function. However, the proof of this theorem is nonconstructive and it is still not known how to actually construct packings that are this dense.

See also Hermite Constants, Hypersphere Packing

References

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 14-16, 1993.