{
  "abstract": "For any non-zero periodic function f : Z → C of period N ⩾ 1 and any non-zero algebraic number z_0 that doesn't lie on a half-line [e^2iℓπ/N , e^2iℓπ/N ∞[, 0 ⩽ ℓ ⩽ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z_0 )-vector space spanned by the numbers L(f, i, z_0 ) : sum_{m=1}^∞ f(m)z_0^m/m^i. This generalizes a result Fischler proved in 2026, corresponding to f identically equal to 1 and |z_0| ⩽ 1: in this case, the numbers L(f, i, z_0 ) = Li_i (z_0 ) are polylogarithm values. Except for a finite number of cuts in the complex plane, our result still holds when |z_0 | > 1, that is in a domain where the series definition above for the numbers L(f, i, z_0 ) doesn't converge anymore, and we make sense of these numbers through analytic continuation. To obtain this result, we construct linear combinations of the numbers L(f, i, z_0 ) using a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on an integral representation of the polylogarithm functions and on a \"Shidlovskii lemma\".",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09774",
  "author_total": 1,
  "authors": [
    {
      "name": "Ludovic Mistiaen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/28b25y98qz025.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09774",
  "fulltext_url": null,
  "id": "28b25y98qz025",
  "issns": [],
  "landing_url": "https://hal.science/hal-05766301v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
  "licence_label": "CC BY-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-nt"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7221007225",
  "pdf_url": "https://hal.science/hal-05766301/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/28b25y98qz025.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "hal-05766301",
      "source": "hal",
      "url": "https://hal.science/hal-05766301v1"
    },
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence",
  "title_text": "Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/28b25y98qz025",
  "venue": "HAL",
  "versions": []
}