A conditional bound $Λ\le0.13963$ for the de Bruijn--Newman constant

Source abstract

Assuming that $ζ(s)$ has no nontrivial zero with $\mathrm{Re}\, s > 7/8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $Λ\le 13963/100000 = 0.13963$ 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 $3 \le \mathrm{Im}\, z \le 6$ 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.