{
  "abstract": "FINDING: Mellin transform bridges Dirichlet series and modular forms via integral representations, with Apéry numbers emerging as special values of modular forms' L-functions. | MATH: Mellin transform \\( \\mathcal{M}[f](s) = \\int_0^\\infty f(x) x^{s-1} dx \\); for modular form \\( f(\\tau) = \\sum a_n q^n \\), \\( L(f,s) = \\mathcal{M}[f(iy)](s) \\) up to gamma factors; Apéry numbers \\( A_n = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n+k}{k}^2 \\) satisfy \\( \\sum A_n q^n = \\) modular form of weight 4 (for \\( \\zeta(3) \\) case); Riemann zeta \\( \\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{x^{s-1}}{e^x-1} dx \\). | CONNECTION: The critical line \\( \\Re(s)=1/2 \\) for zeta — the value 0.5 is not directly in the golden ratio family, but the functional equation \\( s \\leftrightarrow 1-s \\) has fixed point \\( s=1/2 \\), mirroring the self-dual symmetry of \\( \\phi \\) (since \\( \\phi-1 = 1/\\phi \\)). The modular group \\( SL(2,\\mathbb{Z}) \\) has cusp at \\( \\infty \\) and fundamental domain with angles \\( \\pi/3, \\pi\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dqs2wa24f977w.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23409887",
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  "id": "dqs2wa24f977w",
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  "language": "eng",
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  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dqs2wa24f977w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:24Z",
      "native_id": "23409887",
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  "title": "Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research",
  "title_text": "Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:21Z",
  "url": "https://research.draconitic.com/paper/dqs2wa24f977w",
  "venue": "Zenodo",
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