Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 47 min ago
PreprintCC BY 4.0arXiv

On minima of theta and Epstein zeta functions in dimension four

  • Senping Luo
  • Juncheng Wei
Open at sourcePDFDOICite

Abstract

Let the theta and Epstein zeta functions be Θ(α,L)=∑v∈ Le−πα|v|2 for α>0 and E(L,s)=∑v∈ L∖{0}|v|−2s for s>2, respectively. We consider full-rank lattices L⊂ℝ4. Let the covolume of L be one, and let 𝒟4=2−1/4D4 be the root lattice D4 rescaled to covolume one. We prove that, up to orthogonal transformations, nonumber arg mincovol(L)=1Θ(α,L)=𝒟4 if α>0, and this implies that nonumber arg mincovol(L)=1E(L,s)=𝒟4 if s>2. 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 𝒟4. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\cite{End1978}, 1978) and Shushbaev (\cite{Shu1978}, 1978).

Related

  1. Computation of anisotropic singular sums from high-order derivatives of Epstein zeta functions
    Andreas A. Buchheit, Jonathan K. Busse
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryNumerical analysis & scientific computingarXiv
  2. Local minima of the empirical risk in high dimension: General theorems and convex examples
    kiana Asgari, Andrea Montanari A, Basil Saeed
    The Annals of StatisticsPeer-reviewedLicence not statedFull text14 citedStatisticsOpenAlex
  3. Minima of geodesic length functions for non-uniform filling systems
    Subash Chandra Behera, Shiv Parsad
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  4. Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions
    Dandan Chen, Rong Chen, Frank G. Garvan
    The Ramanujan JournalPeer-reviewedLicence not statedFull text5 citedAlgebra & number theoryOpenAlex
  5. Representation zeta functions of self-similar branched groups
    Laurent Bartholdi
    Groups, Geometry, and DynamicsPeer-reviewedLicence not statedFull text5 citedAlgebra & number theoryHAL

Cite

BibTeXRIS
BibTeX
@misc{luo2026minima,
  title = {{On minima of theta and Epstein zeta functions in dimension four}},
  author = {Senping Luo and Juncheng Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37615},
  eprint = {2609.37615},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37615},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On minima of theta and Epstein zeta functions in dimension four
AU  - Senping Luo
AU  - Juncheng Wei
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37615
UR  - https://arxiv.org/abs/2609.37615
AB  - Let the theta and Epstein zeta functions be Θ(α,L)=∑_v∈ Le^−πα|v|^2 for α>0 and E(L,s)=∑_v∈ L∖{0}|v|^−2s for s>2, respectively. We consider full-rank lattices L⊂ℝ^4. Let the covolume of L be one, and let 𝒟_4=2^−1/4D_4 be the root lattice D_4 rescaled to covolume one. We prove that, up to orthogonal transformations, nonumber arg min_covol(L)=1Θ(α,L)=𝒟_4 if α>0, and this implies that nonumber arg min_covol(L)=1E(L,s)=𝒟_4 if s>2. 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 𝒟_4. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\cite{End1978}, 1978) and Shushbaev (\cite{Shu1978}, 1978).
ER  -