On minima of theta and Epstein zeta functions in dimension four

Let the theta and Epstein zeta functions be $\mathrm{\Theta }(\alpha ,L)={\sum }_{v\in L}{e}^{-\pi \alpha \mathrm{\mid }v{\mathrm{\mid }}^2}$ for $\alpha >0$ and $E(L,s)={\sum }_{v\in L\setminus \{0\}}\mathrm{\mid }v{\mathrm{\mid }}^{-2s}$ for $s>2$, respectively. We consider full-rank lattices $L\subset {\mathbb{R}}^4$. Let the covolume of $L$ be one, and let ${\mathcal{D}}_4=2^{-1\mathrm{/}4}{D}_4$ 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).