{
  "abstract": "The discovery that the generalized Riemann functional equation from the literature can be extended to real $b \\ge 0$, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter $(b)$, led to the search for a similar relation for the Lerch $Φ$ function, $Φ(e^z,k,b)$, where a new formula is used as a starting point and the symmetries with respect to $b$ are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to $b>1$, with the aid of a new self-contained Lerch $Φ$ formula, making the relation hold for all real $b$. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch $Φ$ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10609",
  "author_total": 1,
  "authors": [
    {
      "name": "José Risomar Sousa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2pntjp14033h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10609",
  "fulltext_url": null,
  "id": "c2pntjp14033h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10609",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221093247",
  "pdf_url": "https://arxiv.org/pdf/2610.10609",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2pntjp14033h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10609",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10609"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Generalization of the Riemann Functional Equation with Three Variables",
  "title_text": "A Generalization of the Riemann Functional Equation with Three Variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/c2pntjp14033h",
  "venue": "arXiv",
  "versions": []
}