On Sarnak--Strömbergsson conjecture

Let $\Th(α,L)=\sum_{v\in L}e^{-πα|v|^2}$ for $α>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}|v|^{-2s}$ for $s>3/2$ 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{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;&\text{if}\;\; α>1, \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{if}\;\; α<1, \boldsymbol{\mathrm{FCC}}\;\;\mathrm{or}\;\; \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice},\;\;&\text{if}\;\;α=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\text{if}\;\; s>3/2. \end{equation} Therefore, we prove the Sarnak--Strömbergsson conjecture \cite[Inequalities (43)--(44), Section 5]{SS}.