{
  "abstract": "Let the theta and Epstein zeta functions be $Θ(α,L)=\\sum_{v\\in L}e^{-πα|v|^2}$ for $α>0$ and $E(L,s)=\\sum_{v\\in L\\setminus\\{0\\}}{|v|^{-2s}}$ for $s>2$, respectively. We consider full-rank lattices $L\\subset\\mathbb R^4$. Let the covolume of $L$ be one, and let $\\mathcal D_4=2^{-1/4}D_4$ be the root lattice $D_4$ rescaled to covolume one. We prove that, up to orthogonal transformations, \\begin{equation}\\nonumber arg\\,min_{covol(L)=1}Θ(α,L)=\\mathcal D_4 \\qquad\\text{if}\\quad α>0, \\end{equation} and this implies that \\begin{equation}\\nonumber arg\\,min_{covol(L)=1}E(L,s)=\\mathcal D_4 \\qquad\\text{if}\\quad s>2. \\end{equation} 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 $\\mathcal D_4$. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\\cite{End1978}, 1978) and Shushbaev (\\cite{Shu1978}, 1978).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37615",
  "author_total": 2,
  "authors": [
    {
      "name": "Senping Luo"
    },
    {
      "name": "Juncheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmapwrrzvg9hj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37615",
  "fulltext_url": null,
  "id": "dmapwrrzvg9hj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37615",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
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  ],
  "openalex": "W7214891586",
  "pdf_url": "https://arxiv.org/pdf/2609.37615",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmapwrrzvg9hj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.37615",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37615"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "On minima of theta and Epstein zeta functions in dimension four",
  "title_text": "On minima of theta and Epstein zeta functions in dimension four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/dmapwrrzvg9hj",
  "venue": "arXiv",
  "versions": []
}