{
  "abstract": "Let $P\\in \\mathbb Z[X]\\setminus\\{0\\}$ be of degree $\\delta\\ge 1$ and usual height $H\\ge 1$, and let $\\alpha\\in \\overline{\\mathbb Q}^*$ be of degree $d\\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\\varepsilon>0$, there exists $c>0$ such that $\\vert P(J_0'(\\alpha)/J_0(\\alpha))\\vert>c/H^{4d^2\\delta+\\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2\\delta$ can be replaced by a smaller (explicit) quantity $\\mu(d,\\delta)\\le 4d^2\\delta-2d\\delta-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^\\alpha$ for any $\\alpha\\in \\overline{\\mathbb Q}^*$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17641",
  "author_total": 2,
  "authors": [
    {
      "name": "Stéphane Fischler"
    },
    {
      "name": "Tanguy Rivoal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9024xjxm8x3h0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17641",
  "fulltext_url": null,
  "id": "9024xjxm8x3h0",
  "issns": [],
  "landing_url": "https://hal.science/hal-05743630v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
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      "hal",
      "0.math"
    ],
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      "hal",
      "1.math.math-nt"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05743630/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9024xjxm8x3h0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:31Z",
      "native_id": "hal-05743630",
      "source": "hal",
      "url": "https://hal.science/hal-05743630v1"
    },
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments",
  "title_text": "Transcendence measure for the values of the logarithmic derivative of the Bessel function J_0 at algebraic arguments",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/9024xjxm8x3h0",
  "venue": "HAL",
  "versions": []
}