{
  "abstract": "For $n\\geq 0$, Stirling-Ramanujan constants $S_n$ are the Ramanujan summation of the divergent series $\\sum_{k\\geq 1} k^n\\log k$. These constants are exponential periods over the exponential base field $\\mathbb{E}=\\mathbb{Q}(t,e^{-t})$. We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials $P$ and $Q$, with $\\Re Q(s)>0$ for $\\Re s >0$, the constant $S(P,Q)$ is the Ramanujan summation of the series $\\sum_{k\\geq 1} P(k)\\log Q(k)$. They are exponential periods over $\\mathbb{K}_P((ω)) (t,e^{-t}, (e^{-ωt}))$ where $\\mathbb{K}_P$ is the field of definition of $P$ and $(ω)$ are the zeros of $Q$. These constants are related to the derivatives $ζ'_H(-n,w)$ of Hurwitz zeta function at negative integers, which we prove are exponential periods over $\\mathbb{E} \\left ( w,e^{-wt}\\right )$. They can be expressed using Bendersky Gamma functions $\\hat Γ_n$ and the values $\\log \\hat Γ_n(w)$ for $\\Re w >0$ are exponential periods over $\\mathbb{E} \\left ( w,e^{-wt}\\right )$. The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over $\\mathbb{E}$. The values $ζ(2n+1)/π^{2n}$ are exponential periods over $\\mathbb{E}$. For a periodic function $χ:\\mathbb{Z}\\to \\mathbb{C}$, we extend Ramanujan summation to the twisted series $\\sum_{k\\geq 1} χ(k) P(k)\\log Q(k)$ and define General Twisted Stirling-Ramanujan constants $S_χ(P,Q)$. We derive integral formulas proving that they are exponential periods over $\\mathbb{E}(χ)=\\mathbb{Q}(χ)(t,e^{-t})$. For $n\\geq 0$, $L_χ'(-n)$ and $L_χ(n+1)/π^n$ are given in terms of these constants and are exponential periods over $\\mathbb{E}(χ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22113",
  "author_total": 2,
  "authors": [
    {
      "name": "Mounir Hajli"
    },
    {
      "name": "Ricardo Perez-Marco"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bk3aw969rpyzn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22113",
  "fulltext_url": null,
  "id": "bk3aw969rpyzn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22113",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7204207525",
  "pdf_url": "https://arxiv.org/pdf/2608.22113",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bk3aw969rpyzn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2608.22113",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22113"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "General Stirling-Ramanujan Constants are exponential periods and applications",
  "title_text": "General Stirling-Ramanujan Constants are exponential periods and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/bk3aw969rpyzn",
  "venue": "arXiv",
  "versions": []
}