论文标题
爱泼斯坦Zeta功能的紧密反向Minkowski不平等
A Tight Reverse Minkowski Inequality for the Epstein Zeta Function
论文作者
论文摘要
我们证明，如果$ \ MATHCAL {L} \ subset \ Mathbb {r}^n $是一个晶格，以至于所有Sublattices $ \ Mathcal {l} \ [ \ sum _ {\ supt {\ Mathbf {y} \ in \ Mathcal {l} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ neq \ neq \ MathBf0}}}（\ | \ | \ | \ | \ | \ Mathbf {y} \ in \ Mathbb {z}^n \\\ m athbf {z} \ neq \ neq \ mathbf {0}}}}}}}}}}（\ | \ | \ | \ Mathbf {z} \ |^2+q）^{ - S} \]对于所有$ s> n/2 $和所有$ 0 \ leq q \ leq（2S-n）/（n+2）$，当且仅当$ \ mathcal {l} $均为等值时，均为$ \ \ m arthbb {z}^n $。
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 $\mathbb{Z}^n$.
