Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 42 min ago
PreprintCC BY-NC-ND 4.0HAL

Transcendence measure for the values of the logarithmic derivative of the Bessel function J0 at algebraic arguments

  • Stéphane Fischler
  • Tanguy Rivoal
Open at sourcePDFDOICite

Abstract

Let P∈ ℤ[X]∖{0} be of degree δ≥ 1 and usual height H≥ 1, and let α∈ ℚ̅* be of degree d≥ 2. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any ε>0, there exists c>0 such that | P(J0'(α)/J0(α))|>c/H4d2δ+ε where J0 is the Bessel function. In this paper, we prove that the exponent 4d2δ can be replaced by a smaller (explicit) quantity μ(d,δ)≤ 4d2δ−2dδ−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 eα for any α∈ ℚ̅*.

Related

  1. Ω-results for the logarithmic derivative of ζ(s) on vertical homogeneous progressions
    Shengbo Zhao
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Hyper-derivative Algebraic Geometry Codes via Local Expansions
    Xiaofeng Liu, Hengfeng Liu, Jun Zhang +2
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  3. Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
    Javier Segura, Soichiro Suzuki
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Oka complements in algebraic tori and their logarithmic covers
    Yun-Heng Du, Bin Guo, Peng-Chao Wang +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
    Cameron MacMahon
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fischler2026transcendence,
  title = {{Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments}},
  author = {Stéphane Fischler and Tanguy Rivoal},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17641},
  eprint = {2609.17641},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05743630v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Transcendence measure for the values of the logarithmic derivative of the Bessel function J_0 at algebraic arguments
AU  - Stéphane Fischler
AU  - Tanguy Rivoal
PY  - 2026
DA  - 2026/09/09
PB  - HAL
DO  - 10.48550/arxiv.2609.17641
UR  - https://hal.science/hal-05743630v1
AB  - Let P∈ ℤ[X]∖{0} be of degree δ≥ 1 and usual height H≥ 1, and let α∈ ℚ̅^* be of degree d≥ 2. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any ε>0, there exists c>0 such that | P(J_0'(α)/J_0(α))|>c/H^4d^2δ+ε where J_0 is the Bessel function. In this paper, we prove that the exponent 4d^2δ can be replaced by a smaller (explicit) quantity μ(d,δ)≤ 4d^2δ−2dδ−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^α for any α∈ ℚ̅^*.
ER  -