{
  "abstract": "We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\\ldots,x_m,u) = \\prod_{j=1}^m(1+x_j) + u\\prod_{j=1}^m(1-x_j), \\qquad m\\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\\mathrm m_{\\mathbb D}(P_m)$. We prove that, for every fixed $m$, the value $π^m\\mathrm m_{\\mathbb D}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of $\\operatorname{Li}_4(1/2)$, $ζ(3)$, Catalan's constant, $π$, and $\\log2$. We also derive an explicit three-term asymptotic expansion for $\\mathrm m_{\\mathbb D}(P_m)$ as $m\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01951",
  "author_total": 2,
  "authors": [
    {
      "name": "Quanyu Tang"
    },
    {
      "name": "Shu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6t6246cgf57av.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01951",
  "fulltext_url": null,
  "id": "6t6246cgf57av",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01951",
  "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": "W7172420610",
  "pdf_url": "https://arxiv.org/pdf/2608.01951",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6t6246cgf57av.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2608.01951",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01951"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Integral Representations and Asymptotics for a Family of Areal Mahler Measures",
  "title_text": "Integral Representations and Asymptotics for a Family of Areal Mahler Measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/6t6246cgf57av",
  "venue": "arXiv",
  "versions": []
}