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A computer-assisted lower bound for Landau's constant

  • Frank Wikström
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Abstract

We prove the lower bounds B∞>0.51, L>0.51, where B∞ is the locally univalent Bloch constant and L is Landau's constant, improving the bound 1/2+2·10−8 of Chen and Shiba. The argument passes to the logarithm g=log f' of a normalized locally univalent Bloch function, where the Bloch condition becomes a one-sided linear constraint on Re g; positive Toeplitz moment matrices then provide a finite-dimensional outer relaxation for the low logarithmic coefficients, and any non-negative dual vector for a linear program over that relaxation certifies a bound. Floating-point linear programming is used only to discover such dual vectors; the same duality also certifies the second-derivative bounds that control the Taylor allowances and the coefficient caps that contract the cover. An independent arbitrary-precision interval-arithmetic program reconstructs every witness and checks a finite cover of the normalized coefficient body by $1179$ boxes, the smallest certified radius being $0.51003$. All numerical premises were verified with Arb at $80$ decimal digits.

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Cite

BibTeXRIS
BibTeX
@misc{wikstrom2026computerassisted,
  title = {{A computer-assisted lower bound for Landau's constant}},
  author = {Frank Wikström},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23236},
  eprint = {2609.23236},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23236},
}
RIS
TY  - UNPB
TI  - A computer-assisted lower bound for Landau's constant
AU  - Frank Wikström
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23236
UR  - https://arxiv.org/abs/2609.23236
AB  - We prove the lower bounds B_∞>0.51, L>0.51, where B_∞ is the locally univalent Bloch constant and L is Landau's constant, improving the bound 1/2+2·10^−8 of Chen and Shiba. The argument passes to the logarithm g=log f' of a normalized locally univalent Bloch function, where the Bloch condition becomes a one-sided linear constraint on Re g; positive Toeplitz moment matrices then provide a finite-dimensional outer relaxation for the low logarithmic coefficients, and any non-negative dual vector for a linear program over that relaxation certifies a bound. Floating-point linear programming is used only to discover such dual vectors; the same duality also certifies the second-derivative bounds that control the Taylor allowances and the coefficient caps that contract the cover. An independent arbitrary-precision interval-arithmetic program reconstructs every witness and checks a finite cover of the normalized coefficient body by $1179$ boxes, the smallest certified radius being $0.51003$. All numerical premises were verified with Arb at $80$ decimal digits.
ER  -