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 \(\det(\mathcal{L}') \geq 1\) for all sublattices \(\mathcal{L}' \subseteq \mathcal{L}\), 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 \(0 \leq q \leq (2s-n)/(n+2)\), with equality if and only if \(\mathcal{L}\) is isomorphic to \(\mathbb{Z}^n\).