{
  "abstract": "The secondary zeta function is defined by $$ Z(s)=\\sum_{γ>0}\\frac{1}{γ^{s}}, $$ where $\\tfrac12+iγ$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $ξ$-function. Define $$ L(x)=\\log\\left(\\frac{ξ(x+\\tfrac12)}{ξ(\\tfrac12)}\\right), \\qquad \\mathcal{E}=x\\frac{\\mathrm{d}}{\\mathrm{d}x}, \\qquad \\mathcal{D}_k=\\mathcal{E}\\prod_{j=1}^{k}(\\mathcal{E}-2j). $$ Then, for $k\\geq1$, $$ Z(2k)=\\frac{(-1)^k}{2^k(k-1)!}\\int_{0}^{\\infty}\\frac{\\mathcal{D}_k L(x)}{x^{2k+1}}\\,\\mathrm{d}x $$ and $$ Z(2k+1)=\\frac{(-1)^k}{π\\,(2k-1)!!}\\int_{0}^{\\infty}\\frac{\\mathcal{D}_k L(x)}{x^{2k+2}}\\,\\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\\leq k\\leq 7$, both in terms of $L(x)$ and in terms of $ξ(x+\\tfrac12)$ for reference. We also derive a limit formula for $γ_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19175",
  "author_total": 1,
  "authors": [
    {
      "name": "Artur Kawalec"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d23feghaxs1xm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19175",
  "fulltext_url": null,
  "id": "d23feghaxs1xm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19175",
  "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": "W7222277610",
  "pdf_url": "https://arxiv.org/pdf/2610.19175",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d23feghaxs1xm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19175",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19175"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Even and Odd Integer Values of the Secondary Zeta Function",
  "title_text": "Even and Odd Integer Values of the Secondary Zeta Function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/d23feghaxs1xm",
  "venue": "arXiv",
  "versions": []
}