HEXAGONbeta

How long can an off-line zero survive the de Bruijn-Newman heat flow? An explicit unconditional lower bound for Lambda

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Pedro Caceres ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

Let Λ\Lambda be the de Bruijn-Newman constant. The Riemann hypothesis (RH) is equivalent to Λ≤0\Lambda\le 0, and Rodgers and Tao proved Λ≥0\Lambda\ge 0. We ask how small Λ\Lambda can be if ζ\zeta has a zero 1/2+δ+iγ1/2+\delta+i\gamma with δ>0\delta>0. We use de Bruijn's strip, the zero dynamics of the Polymath project, and an explicit local zero count for the heat-flow deformations HτH_\tau that we prove here. With these we show unconditionally that Λ≥δ2/(70log⁡γlog⁡log⁡γ)\Lambda \ge \delta^2/(70\log\gamma\log\log\gamma). Equivalently, every zero β+iγ\beta+i\gamma of ζ\zeta with γ≥3⋅1012\gamma\ge 3\cdot 10^{12} satisfies ∣β−1/2∣≤(70 Λlog⁡γlog⁡log⁡γ)1/2|\beta-1/2|\le (70\,\Lambda\log\gamma\log\log\gamma)^{1/2}; below that height RH has been verified. We also give the bound as an explicit formula at each height, which improves slowly with γ\gamma. The proof follows a weighted sum ∑ψ(Re z)(Im z)p\sum\psi(\mathrm{Re}\,z)(\mathrm{Im}\,z)^p over all zeros of HτH_\tau in the upper half-plane, which is insensitive to collisions and to other off-line zeros. For a single pair, the imaginary part obeys an exact equation whose interaction term at τ=0\tau=0 is the energy slope ∂σlog⁡∣ξ∣\partial_\sigma\log|\xi| of the other zeros. Under local isolation and regular spacing we prove a sharper bound, linear in δ\delta. Simulations with the zeros up to height 10410^4 agree with both bounds. We do not prove RH.

Provenance statement

The research question and the program behind it are the author’s. The question is how small the de Bruijn–Newman constant can be if zeta has an off-line zero, and it grew out of the author’s search for a “distance to the critical line” with rigidity. The link with the cross-energy / energy-slope program (introduced by the author in 2017) is also the author’s. The author used Claude (Anthropic), an AI assistant, as a research and writing aid. Claude derived and checked the proofs, computed the explicit constants, ran the numerical simulations in Python, searched the literature, and drafted the LaTeX text and figures. A separate AI review agent with no part in the drafting gave two rounds of critical refereeing. The author directed the work, checked the results and takes full responsibility for the content. No AI system is an author. The paper does not claim a proof of the Riemann hypothesis.

Tools used

Anthropic
Claude OpusVersion 5.5
Python
PythonVersion 3.13

References

  1. P. Caceres, The cross-energy of Dirichlet partial sums and the zeros of the Riemann zeta function , consolidated paper, v2 (2026); v1: Zenodo, doi:10.5281/zenodo.23243641.DOI
  2. N. G. de Bruijn, The roots of trigonometric integrals, Duke Math.\ J. 17 (1950), 197--226.
  3. C. M. Newman, Fourier transforms with only real zeros, Proc.\ Amer.\ Math.\ Soc. 61 (1976), 245--251.
  4. F. W. J. Olver, Asymptotics and Special Functions , Academic Press, 1974.
  5. M. Planat, Collective contraction in the de Bruijn--Newman heat flow: off-zero logarithmic derivatives and certified barrier refinements, preprint, arXiv:2609.37164 (2026).arXiv
  6. D. J. Platt, T. S. Trudgian, The Riemann hypothesis is true up to $3ḑot10^ 12 $, Bull.\ London Math.\ Soc. 53 (2021), 792--797.
  7. D. H. J. Polymath, Effective approximation of heat flow evolution of the Riemann $ $ function, and a new upper bound for the de Bruijn--Newman constant, Res.\ Math.\ Sci. 6 (2019), 31.
  8. B. Rodgers, T. Tao, The de Bruijn--Newman constant is non-negative, Forum Math.\ Pi 8 (2020), e6.
  9. T. Tao, Polymath15, first thread: computing HtH_t, asymptotics, and dynamics of zeroes, blog post, 27 January 2018.
  10. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., revised by D. R. Heath-Brown, Oxford, 1986.

Version history

  1. v1Submitted by Pedro CaceresInitial depositCurrentOct 12, 2026