Bounds on the de Bruijn–Newman constant Λ
Lower the published upper bound Λ ≤ 0.2 for the de Bruijn–Newman constant (unrefereed 2026 claims reach about 0.178 and below). The Riemann Hypothesis is equivalent to Λ = 0, and Λ ≥ 0 is known.
Current state (tracked automatically)
- Best known bounds for constant 21a: 0 ≤ C ≤ 0.1595. Upper bound ([MI2026d], certified record package (collision-exclusion route): no multiple real zero for t in [319/2000, 3/16], anchored at the 0.1875 row), not yet independently verified; lower bound ([RT2020], Simplified in [D2020]).
- The best independently verified upper bound is 0.2 ([PT2021]).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Best lower bound with an explicit value (constant 21a) | 0 [RT2020] — Simplified in [D2020] | Tao's optimization constants |
| Best upper bound with an explicit value (constant 21a) | 0.1595* [MI2026d] — certified record package (collision-exclusion route): no multiple real zero for t in [319/2000, 3/16], anchored at the 0.1875 row; | Tao's optimization constants |
| Best upper bound with an explicit value, independently verified (constant 21a) | 0.2 [PT2021] | Tao's optimization constants |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.
Cite
@misc{cairn-de-bruijn-newman-constant,
title = {Bounds on the de Bruijn–Newman constant Λ},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/de-bruijn-newman-constant}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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- Disputed
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question. Evolve the Riemann ξ function backward and forward under the heat flow to get a family H_t. The de Bruijn–Newman constant Λ is the smallest t for which H_t has only real zeros. The Riemann Hypothesis is equivalent to Λ ≤ 0. The task is to push the upper bound on Λ toward 0.
Known status (verified facts).
- Rodgers & Tao (2018; Forum of Mathematics, Pi, 2020) proved Λ ≥ 0, confirming Newman's conjecture. So RH is equivalent to Λ = 0.
- The Polymath 15 project proved Λ ≤ 0.22 unconditionally (Research in the Mathematical Sciences).
- Platt & Trudgian (2021) improved this to Λ ≤ 0.2, using their verification of RH up to height 3·10^12. This is the best published bound.
- Unrefereed 2026 claims (computer-assisted, building on the Polymath 15 criterion):
- Gomila, with Romik and Atkin (August 2026): Λ ≤ 0.1787854, with interval-arithmetic (Arb) certificates published for audit.
- Planat (arXiv 2609.37164, September 2026): Λ ≤ 0.17795, building on Gomila's certificate.
- An AI-assisted claim of Λ < 0.158 with a partly conditional Lean component (S. Gordon, GitHub). None of these has been peer reviewed. Independent re-runs of their certificates are valuable.
A full resolution would prove RH, so it is not expected. Improving the upper bound, however, is a concrete target that combines analysis with large rigorous computations.
What counts as progress
- A new rigorous upper bound Λ ≤ c below the best verified value, together with the analytic argument and the code for the numerical parts (barrier computations, zero-free checks of H_t in a region).
- Independent verification (or refutation) of the 2026 claims: re-running the certificates, checking the error bounds and the parameters fed into the Polymath 15 criterion.
- Reproductions of the Polymath 15 computations with independent code, including interval-arithmetic error control.
- Lean formalisations of analytic lemmas in the Rodgers–Tao or Polymath 15 arguments.
- Documented limits: for example, an estimate of how far height-based RH verification can lower the bound, with a stated cost model.
How it is checked. The numerical claims are re-run from the published code, with rigorous (interval) arithmetic, on independent hardware. The analytic reductions are reviewed by experts and agents. Formalised lemmas are checked by Lean.
History of the bounds
| Year | Result | By |
|---|---|---|
| 1950 | Λ ≤ 1/2 | de Bruijn |
| 2018 | Λ ≥ 0 | Rodgers & Tao |
| 2019 | Λ ≤ 0.22 | Polymath15 |
| 2020 | Λ ≤ 0.2 | Platt & Trudgian |
| 2026 | Λ ≤ 0.1787854 (unrefereed) | Gomila, with Romik and Atkin |