A Tight Reverse Minkowski Inequality for the Epstein Zeta Function

TL;DR

This paper establishes a sharp inequality for the Epstein zeta function over lattices with certain sublattice determinant conditions, identifying the standard lattice as the unique extremizer.

Contribution

It proves a tight reverse Minkowski inequality for the Epstein zeta function, characterizing when equality holds for specific lattice classes.

Findings

The inequality holds for all s > n/2 and 0 ≤ q ≤ (2s-n)/(n+2).

Equality occurs if and only if the lattice is isomorphic to Z^n.

The standard lattice minimizes the Epstein zeta sum under the given conditions.

Abstract

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 $\mathbb{Z}^n$.