Sarnak–Strömbergsson conjecture

Let $L\subset\mathbb{R}^3$ range over Bravais lattices of covolume $1$ . Define $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha\lvert v\rvert^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}\lvert v\rvert^{-2s}$ for $s>\frac32$ . Up to orthogonal transformations, the conjecture asserts that $\operatorname*{argmin}_{\operatorname{covol}(L)=1}\Theta(\alpha,L)$ is the FCC lattice when $\alpha>1$ , the BCC lattice when $\alpha<1$ , and consists of the FCC and BCC lattices when $\alpha=1$ . It also asserts that $\operatorname*{argmin}_{\operatorname{covol}(L)=1}E(L,s)$ is the FCC lattice for every $s>\frac32$ .

References

Primary source

arXiv

Additional references

On Sarnak--Strömbergsson conjecture — arXiv — Senping Luo, Juncheng Wei

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture, but the claim has not been independently checked.

The conjecture predicts which of the two three-dimensional crystal lattices, $\mathrm{FCC}$ or $\mathrm{BCC}$ , minimizes theta and Epstein zeta energies in the relevant parameter ranges. It is attributed to Sarnak and Strömbergsson.

Known results

A 2016 paper records the global minimization statements as open for unit-density three-dimensional Bravais lattices.
Earlier work proves local optimality of $\mathrm{FCC}$ and $\mathrm{BCC}$ in suitable parameter regimes.
$\mathbb{Z}^3$ is shown to be a saddle point rather than a global minimizer.

September 2026 claimed proof

A preprint by Senping Luo and Juncheng Wei claims the $\mathrm{FCC}/\mathrm{BCC}$ phase split for the theta function and $\mathrm{FCC}$ optimality for the Epstein zeta function for all $s>3/2$ among unit-covolume three-dimensional lattices. If correct, this resolves the stated conjecture.

Current status (as of September 2026): A preprint claims the full three-dimensional conjecture, but no independent verification is recorded, so the result remains unconfirmed.

Sources

No solutions have been posted yet.