A Tight Reverse Minkowski Inequality for the Epstein Zeta Function

Abstract

We prove that if $\mathcal{L}\subset {\mathbb{R}}^n$ is a lattice such that $\det ({\mathcal{L}}^{\prime })\ge 1$ for all sublattices ${\mathcal{L}}^{\prime }\subseteq \mathcal{L}$, then \[ \sum_{\substack{\mathbf{y}\in\mathcal{L}\\\mathbf{y}\neq\mathbf0}} (\|\mathbf{y}\|^2+q)^{-s} \leq \sum_{\substack{\mathbf{z} \in \mathbb{Z}^n\\\mathbf{z}\neq\mathbf{0}}} (\|\mathbf{z}\|^2+q)^{-s} \] for all $s>n/2$ and all $0\le q\le (2s-n)/(n+2)$, with equality if and only if $\mathcal{L}$ is isomorphic to ${\mathbb{Z}}^n$.