{
  "abstract": "An explicit formula for the prime-counting function $π(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $π(x)=R(x)-\\sum_ρR(x^ρ)$, where the sum runs over all zeros $ρ$ of the Riemann $ζ$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $ΣR_T(x):=\\sum_{0<|\\Im m(ρ)|\\le T} R(x^ρ)$ converge as ${T\\to\\infty}$. Writing $Θ:=\\sup\\{\\Re e(ρ):\\ ζ(ρ)=0,\\ 0<\\Re e(ρ)<1\\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $θ<Θ$, the sums $ΣR_T(x)$ are not $O(T^θ)$. As a consequence, $\\limsup_{T\\to\\infty}|ΣR_T(x)|=\\infty$ and $\\sum_ρR(x^ρ)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result -- and hence divergence -- for the contribution of the trivial zeros to $\\sum_ρR(x^ρ)$. In both cases the exponent $Θ$ is proved to be sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02713",
  "author_total": 1,
  "authors": [
    {
      "name": "Harald Grobner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jqweevqarwpd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02713",
  "fulltext_url": null,
  "id": "9jqweevqarwpd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02713",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7208794464",
  "pdf_url": "https://arxiv.org/pdf/2609.02713",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9jqweevqarwpd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.02713",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02713"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function",
  "title_text": "On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/9jqweevqarwpd",
  "venue": "arXiv",
  "versions": []
}