On Sarnak--Strömbergsson conjecture

Let $\text{Th}(\alpha ,L)={\sum }_{v\in L}{e}^{-\pi \alpha \mathrm{\mid }v{\mathrm{\mid }}^2}$ for $\alpha >0$ and $E(L,s)={\sum }_{v\in L\setminus \{0\}}\mathrm{\mid }v{\mathrm{\mid }}^{-2s}$ for $s>3\mathrm{/}2$ be the theta and Epstein zeta functions associated to the lattice $L$, respectively. We are particularly interested in physically relevant dimension three. Fix the covolume of the lattice $L$ to $1$. Up to an orthogonal transformation, we prove that \begin{equation}\nonumber \argmin_{|L|=1}\Th(α,L)= \begin{cases} \boldsymbol{$\mathrm{F}\mathrm{C}\mathrm{C}$};;$\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$,;;;;;;;;; ;;;;;;&$\text{if}$;; α>1, \boldsymbol{$\mathrm{B}\mathrm{C}\mathrm{C}$};;$\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$, ;;;;;;;;;;;;;;;&$\text{if}$;; α<1, \boldsymbol{$\mathrm{F}\mathrm{C}\mathrm{C}$};;$\mathrm{o}\mathrm{r}$;; \boldsymbol{$\mathrm{B}\mathrm{C}\mathrm{C}$};;$\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$,;;&$\text{if}$;;α=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{$\mathrm{F}\mathrm{C}\mathrm{C}$};;$\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$,;;;;;;;;; ;;;;;;$\text{if}$;; s>3/2. \end{equation} Therefore, we prove the Sarnak–Strömbergsson conjecture \cite[Inequalities (43)–(44), Section 5]{SS}.