{
  "abstract": "We introduce a family of discrete hyperbolic secant distributions on $\\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant $G=\\varpi/π$, where $\\varpi=Γ^2(1/4)/(2\\sqrt{2π})$ is the lemniscate constant. Using the elliptic lambda-star function $λ^*$, we construct scaled versions of these distributions parametrized by $\\sqrt{r}$ and $1/\\sqrt{r}$ for $r \\in \\mathbb{N}$. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the $-2πi$-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\\,g.\\ $\\sum_{k \\in \\mathbb{Z}} \\operatorname{sech}^3(πk) = (G^3+G)/\\sqrt{2}$, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11293",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonard Pleschberger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm3bxt1d8fgyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11293",
  "fulltext_url": null,
  "id": "bm3bxt1d8fgyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11293",
  "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.PR"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212221148",
  "pdf_url": "https://arxiv.org/pdf/2609.11293",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bm3bxt1d8fgyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11293",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11293"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Discrete Hyperbolic Secant Distributions",
  "title_text": "Discrete Hyperbolic Secant Distributions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/bm3bxt1d8fgyr",
  "venue": "arXiv",
  "versions": []
}