A conditional bound Λ≤0.13963 for the de Bruijn--Newman constant
Abstract
Assuming that ζ(s) has no nontrivial zero with Re s > 7/8, as asserted in a recent unrefereed preprint of OpenAI, we obtain Λ≤ 13963/100000 = 0.13963 for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width 1/5. We derive an approximation on 3 ≤ Im z ≤ 6 from its general Riemann-Siegel coefficient bounds. At height 61/20, a fixed cutoff permits two differentiations, and comparison with ζ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon's analytic inputs nor Gomila's new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.
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BibTeX
@misc{hakimoglubrown2026conditional,
title = {{A conditional bound $Λ\le0.13963$ for the de Bruijn--Newman constant}},
author = {Cetin Hakimoglu-Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12803},
eprint = {2610.12803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A conditional bound Λ≤0.13963 for the de Bruijn--Newman constant
AU - Cetin Hakimoglu-Brown
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12803
UR - https://arxiv.org/abs/2610.12803
AB - Assuming that ζ(s) has no nontrivial zero with Re s > 7/8, as asserted in a recent unrefereed preprint of OpenAI, we obtain Λ≤ 13963/100000 = 0.13963 for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width 1/5. We derive an approximation on 3 ≤ Im z ≤ 6 from its general Riemann-Siegel coefficient bounds. At height 61/20, a fixed cutoff permits two differentiations, and comparison with ζ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon's analytic inputs nor Gomila's new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.
ER -