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On Sarnak--Strömbergsson conjecture
Abstract
Let Th(α,L)=∑v∈ Le−πα|v|2 for α>0 and E(L,s)=∑v∈ L∖{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 nonumber argmin|L|=1Th(α,L)= FCC lattice, &if α>1, BCC lattice, &if α<1, FCC or BCC lattice, &if α=1, and nonumber argmin|L|=1E(L,s)= FCC lattice, if s>3/2. Therefore, we prove the Sarnak--Strömbergsson conjecture \cite[Inequalities (43)--(44), Section 5]{SS}.
Related
BibTeX
@misc{luo2026sarnakströmbergsson,
title = {{On Sarnak--Strömbergsson conjecture}},
author = {Senping Luo and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17356},
eprint = {2609.17356},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17356},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Sarnak--Strömbergsson conjecture
AU - Senping Luo
AU - Juncheng Wei
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17356
UR - https://arxiv.org/abs/2609.17356
AB - Let Th(α,L)=∑_v∈ Le^−πα|v|^2 for α>0 and E(L,s)=∑_v∈ L∖{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 nonumber argmin_|L|=1Th(α,L)= FCC lattice, &if α>1, BCC lattice, &if α<1, FCC or BCC lattice, &if α=1, and nonumber argmin_|L|=1E(L,s)= FCC lattice, if s>3/2. Therefore, we prove the Sarnak--Strömbergsson conjecture \cite[Inequalities (43)--(44), Section 5]{SS}.
ER -