{
  "abstract": "One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\\mathcal{Z}(s,w,w^\\prime)$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\\mathcal{Z}(s,w,w^\\prime)$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\\tilde{\\mathcal{Z}}(s,w,w^\\prime)$. In this article, we derive all the Laurent coefficients of $\\tilde{\\mathcal{Z}}(s,w,w^\\prime)$, akin to Ishibashi. These Laurent coefficients involve an interesting function $\\mathfrak{F}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for $\\mathfrak{F}_k^0(x)$. We establish both the two term and the three term functional equation of $\\mathfrak{F}_k^0(x)$, derive the action of the period-like Hecke operator on $\\mathfrak{F}_k^0(x)$ and connect an important integral with $\\mathfrak{F}_k^0(x)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15052",
  "author_total": 2,
  "authors": [
    {
      "name": "Soumyarup Banerjee"
    },
    {
      "name": "Riya Mandal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hvp727g99d95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15052",
  "fulltext_url": null,
  "id": "2hvp727g99d95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15052",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213283517",
  "pdf_url": "https://arxiv.org/pdf/2609.15052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hvp727g99d95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function",
  "title_text": "Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2hvp727g99d95",
  "venue": "arXiv",
  "versions": []
}