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Cross-Energy of Dirichlet Partial Sums. Part 35: How long can an off-line zero survive the de Bruijn-Newman heat flow? An explicit lower bound for Λ\Lambda forced by a zero off the critical line

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

Contributed by Pedro Caceres

Version 2 / 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\le0, and Rodgers and Tao proved Λ≥0\Lambda\ge0. We ask how small Λ\Lambda can be if ζ\zeta has a zero 12+δ+iγ\frac12+\delta+i\gamma with δ>0\delta>0. Using de Bruijn's strip, the zero dynamics of the Polymath project, and an explicit local zero count for the deformations HτH_\tau that we prove here, we show, with no further hypothesis, that such a zero forces \[ \Lambda\ \ge\ \frac{\delta^2}{70\,\log\gamma\,\log\log\gamma}, \] the numerical steps of the proof are certified in ball arithmetic. This is an implication from a hypothetical zero: under RH it says nothing, and the only unconditional lower bound for Λ\Lambda remains Λ≥0\Lambda\ge0. The proof bounds Λ\Lambda, not the survival time of the given zero. For a single pair, the interaction term of the exact equation for its imaginary part is, at τ=0\tau=0, the energy slope ∂σlog⁡∣ξ∣\partial_\sigma\log|\xi| of the other zeros at the zero itself; under local isolation and regular spacing the bound becomes linear in δ\delta. In an exact polynomial model, a pair over a cluster of KK real zeros (KK even) is absorbed at 2δ2/(K+1)2\delta^2/(K+1), and clusters of the size the unconditional zero count allows keep the rate at δ2/log⁡γ\delta^2/\log\gamma in the simulated dynamics; this suggests that a linear bound needs a local count not known even under RH. On the 2121 off-line zeros of the Davenport--Heilbronn function with δ>0.005\delta>0.005 and height below 800800, freezing the energy slope predicts the survival time to within 7%7\%, and we show by a computer-assisted argument that its constant satisfies 0.228<Λf≤3.1250.228<\Lambda_f\le3.125. Finally, we prove a heat-flow Landau formula for fixed nn and 0<t≤120<t\le\frac12: under the flow the memory of a prime decays like p−(t/4)log⁡(γ/2πp)p^{-(t/4)\log(\gamma/2\pi p)} and rotates, while composite numbers acquire a memory governed by a deformed von Mangoldt function Λt\Lambda_t, which is positive for prime powers, for all squarefree nn, and for small tt when nn has two prime factors (numerically in all cases tested); the main term dominates when t4log⁡n<1\frac t4\log n<1.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 40 pages, 6 figures. Ancillary scripts: those of the original part (ancillary zip in the part folder) and, for Section 9, Caceres_CE32_ancillary_front32a.zip. Part 35 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly Part 15. Renumbered 2 Nov 2026. v2 (2 Nov 2026): new Section 9, exact cluster absorption (Hermite model, Proposition 9.1) and the barrier for a linear bound; abstract updated. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

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

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Version history

  1. v1 · 13 pagesSubmitted by Pedro CaceresInitial depositSupersededOct 12, 2026
  2. v2 · 40 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 40 pages, 6 figures. Ancillary scripts: those of the original part (ancillary zip in the part folder) and, for Section 9, Caceres_CE32_ancillary_front32a.zip. Part 35 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly Part 15. Renumbered 2 Nov 2026. v2 (2 Nov 2026): new Section 9, exact cluster absorption (Hermite model, Proposition 9.1) and the barrier for a linear bound; abstract updated. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.CurrentNov 11, 2026