{
  "abstract": "In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $Φ(e^{-t})=\\sum_{n=1}^{\\infty}μ(n)e^{-nt}$, as a function of the values of $ζ(s)$ and $ζ'(s)$ at the odd integers and as a function of the zeros of $ζ(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $Γ(s)$ collide with the trivial zeros of $ζ(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give a criterion for the Riemann hypothesis: the bound $O(x^{-1/2})$ on the transform implies the Riemann hypothesis unconditionally, while the converse direction requires additional hypotheses on the zeros. We also introduce special entire functions related to $ζ(s)$ and show that they admit absolutely convergent closed forms as Möbius-weighted series of Bessel functions of rotated argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.09797",
  "author_total": 1,
  "authors": [
    {
      "name": "Sergey Liflandsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3xvqr22w84qw0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.09797",
  "fulltext_url": null,
  "id": "3xvqr22w84qw0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.09797",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7168298325",
  "pdf_url": "https://arxiv.org/pdf/2607.09797",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3xvqr22w84qw0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2607.09797",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.09797"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis",
  "title_text": "Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/3xvqr22w84qw0",
  "venue": "arXiv",
  "versions": []
}