{
  "abstract": "The derived set of multiple zeta-star values is the half-line $[1,+\\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence and hence admits a normalized Hausdorff--Stieltjes representation. Both the shifted and the unshifted functions, as well as their infinite-depth limits, are shown to be univalent on the half-plane $\\mathrm{Re}\\,z<1$. For finite indices, the representing densities satisfy a strict monotone likelihood-ratio order with respect to the reverse lexicographic order. Together with closure properties of the Hausdorff--Stieltjes class and a limiting quotient argument, this shows that, along the upper semicircle, the argument of each translated curve increases strictly while its modulus decreases strictly; the corresponding lower-semicircle statement follows by conjugation. The same idea gives strict separation of the curves attached to different indices. We then establish a one-to-one correspondence between a natural set of pairs consisting of a point of the unit circle and an infinite index, and the closed half-plane $\\mathrm{Re}\\,w\\geq \\frac12$ with the point $1$ removed. Under a binary coordinate on the index set, this correspondence is a homeomorphism. As an application, a subclass of shifted cyclotomic multiple zeta-star values of all levels form a countable dense subset of the open half-plane $\\mathrm{Re}\\,w>\\frac12$. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane $\\mathrm{Re}\\,w\\geq\\frac12$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03653",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiangtao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jx7exc6ztf38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03653",
  "fulltext_url": null,
  "id": "8jx7exc6ztf38",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03653",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7208719416",
  "pdf_url": "https://arxiv.org/pdf/2609.03653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jx7exc6ztf38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The derived set of multiple star polylogarithms",
  "title_text": "The derived set of multiple star polylogarithms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8jx7exc6ztf38",
  "venue": "arXiv",
  "versions": []
}