On minima of theta and Epstein zeta functions in dimension four

Source abstract

Let the theta and Epstein zeta functions be $Θ(α,L)=\sum_{v\in L}e^{-πα|v|^2}$ for $α>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}{|v|^{-2s}}$ for $s>2$, respectively. We consider full-rank lattices $L\subset\mathbb R^4$. Let the covolume of $L$ be one, and let $\mathcal D_4=2^{-1/4}D_4$ be the root lattice $D_4$ rescaled to covolume one. We prove that, up to orthogonal transformations, $$\begin{equation}\nonumber arg\,min_{covol(L)=1}Θ(α,L)=\mathcal D_4 \qquad\text{if}\quad α>0, \end{equation}$$ and this implies that $$\begin{equation}\nonumber arg\,min_{covol(L)=1}E(L,s)=\mathcal D_4 \qquad\text{if}\quad s>2. \end{equation}$$ This proves that the root lattice minimizes the theta and Epstein zeta functions among all lattices in dimension four, as conjectured by Sarnak-Strömbergsson (\cite{SS}, 2006), who established the local minimality of $\mathcal D_4$. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\cite{End1978}, 1978) and Shushbaev (\cite{Shu1978}, 1978).