A Tight Reverse Minkowski Inequality for the Epstein Zeta Function

We prove that if $\mathcal{L}\subset {\mathbb{R}}^n$ is a lattice such that $\det ({\mathcal{L}}^{\mathrm{\prime }})\ge 1$ for all sublattices $$\mathcal{L}$’ \subseteq $\mathcal{L}$$, then $$\sum _{\begin{array}{c}\mathbf{y}\in \mathcal{L}\\ \mathbf{y}\mathrm{\ne }\mathbf{0}\end{array}}(\mathrm{\parallel }\mathbf{y}{\mathrm{\parallel }}^2+q{)}^{-s}\le \sum _{\begin{array}{c}\mathbf{z}\in {\mathbb{Z}}^n\\ \mathbf{z}\mathrm{\ne }\mathbf{0}\end{array}}(\mathrm{\parallel }\mathbf{z}{\mathrm{\parallel }}^2+q{)}^{-s}$$ for all $s>n\mathrm{/}2$ and all $0\le q\le (2s-n)\mathrm{/}(n+2)$, with equality if and only if $\mathcal{L}$ is isomorphic to ${\mathbb{Z}}^n$.