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Upper bound Λ≤0.1787854\Lambda \le 0.1787854 on the de Bruijn-Newman constant

Let HtH_t be the heat-flow deformation of the Riemann ξ\xi function and Λ\Lambda the de Bruijn-Newman constant: the zeros of HtH_t are all real exactly when t≥Λt \ge \Lambda. The Riemann hypothesis is the statement Λ≤0\Lambda \le 0, and Rodgers and Tao proved Newman's conjecture Λ≥0\Lambda \ge 0 in 2018. How small an upper bound on Λ\Lambda can be proved?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-assisted
Method
Computation
Field
Number theory
Posed by
Nicolaas de Bruijn (1950), whose bound of 1/2 opened the question; lowered by Ki, Kim and Lee (2009), Polymath15 (2019) and Platt-Trudgian's verification height (2020)
Year posed
1950
Years open
76y
Solved
2026-08-20
Model
Claude; ChatGPT and Codex (versions undisclosed)
Vendor
Anthropic; OpenAI
Collaborators
Jude Gomila
Verification
Unreviewed
Publication
Announced
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Λ≤893927/5000000=0.1787854\Lambda \le 893927/5000000 = 0.1787854, below the standing reviewed record of 0.20.2 and the unreviewed 0.18750.1875 of July 2026. The Riemann hypothesis is Λ≤0\Lambda \le 0; Newman's complementary conjecture Λ≥0\Lambda \ge 0, his "quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so", was proved by Rodgers and Tao in 2018 and reproved more directly by Dobner. De Bruijn's original bound was Λ≤1/2\Lambda \le 1/2, and lowering it is one way of measuring progress toward RH. The frontier "Upper bounds for the de Bruijn-Newman constant" tracks these steps and links this entry from its 20 August 2026 row. Tighter bounds and the exact value remain open.

What the AI did

"Generative AI systems, including Anthropic Claude and OpenAI ChatGPT/Codex, were used in repeated, human-directed cycles of mathematical exploration, program development, testing, criticism, and manuscript revision. Jude Gomila selected the arguments and computations included here and accepts responsibility for the mathematical claims, the programs, and the final text."

Verification

Read at the repository on 12 September 2026: the release, the sealed manifest, the referee report and the bibliography. The claim is Polymath15's Theorem 1.2 instantiated at exact parameters X=6000000185827X = 6000000185827, t0=129/800t_0 = 129/800, y02=87677/2500000y_0^2 = 87677/2500000, with fail-closed Arb interval certificates; unconditional in form, using RH only up to the finite Platt-Trudgian height. The referee report is an adversarial AI panel that calls itself no substitute for human review, found no fatal defect and left three items needing human sign-off. Nothing was rerun here: neither the certificates nor the parameter check against Theorem 1.2. The standing reviewed record is 0.2, Polymath15's criterion at Platt-Trudgian's height; the intermediate 0.1875 (Mosaic Intelligence, July 2026) is also unreviewed. Announcement reflects repository-only publication.

Source

FrontierStep on de Bruijn-Newman Λ · 0.1787854 · candidate, under review

Submitted by PluckyWombat559 on

Changelog2 changes

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