{
  "abstract": "We propose a unified framework for Prime Rigidity Theory (PR) by introducing the notions of an \"abstract integer system\" and an \"abstract prime set\", consisting, respectively, of two unbounded sequences $\\mathcal{P}$ and $\\mathcal{N}$ of elements of $\\mathcal{X}$, where $\\mathcal{X}$ denotes the algebra of bounded linear endomorphisms of a complex Banach space $X$, such that every element of $\\mathcal{N}$ can be expressed as a finite product of powers of elements of $\\mathcal{P}$. We define the zeta function $\\zeta_{X,\\mathcal{N}}$ on $\\mathcal{X}\\setminus{I_X}$, where $I_X$ is the identity endomorphism. The scalar restriction $\\zeta_{\\mathbb{C},\\mathcal{N}}$ yields generalized Beurling zeta functions; in particular, $\\zeta_{\\mathbb{C},\\mathbb{N}}$ coincides with the classical Riemann zeta function. We introduce the notions of a \"rigid abstract integer system\" and \"rigid abstract prime set\" in order to preserve the rigid asymmetric structure obtained in the scalar case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "W Oukil"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bet0brk17s8x5.bib",
  "cited_by": 0,
  "doi": null,
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  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
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  "native_subjects": [
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  "pdf_url": "https://hal.science/hal-05682635/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bet0brk17s8x5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:34Z",
      "native_id": "hal-05682635",
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      "url": "https://hal.science/hal-05682635v3"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ct",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ca",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ds",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "From scalar rigidity to abstract prime set and abstract integer system",
  "title_text": "From scalar rigidity to abstract prime set and abstract integer system",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/bet0brk17s8x5",
  "venue": "HAL",
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}