Theorem · version 2
For the classical de Bruijn–Newman heat-flow family, Λ ≥ 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.
Newman’s separate nonnegativity conjecture is proved.
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| The Newman constant satisfies Λ ≥ 0 → Proved Selected source date: | ≥ 0 | Classical heat-flow normalization. |
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Rodgers–Tao · Newman constant is nonnegative · Theorem 1.1; §1 normalization and RH equivalence →
Source dated Jul 3, 2021 · Observed Oct 10, 2026
In this problem · publication v6 · Selected context
Bound · Proved
RH is equivalent to Λ ≤ 0. The proved lower bound makes RH equivalent to Λ = 0; it does not prove that upper bound.
Scope: Rodgers and Tao proved Newman’s nonnegativity conjecture: if RH is true, Λ must be exactly zero.
Limit: Λ ≥ 0 alone allows positive Λ, which would make RH false. The required upper bound Λ ≤ 0 remains open.
A distinct scope, not an additive fraction of the conjecture solved.
The two proved inequalities enclose the same normalized constant in [0, 0.2].
An interval containing zero is not a proof that the constant equals zero.
Proved · version 2
For the classical de Bruijn–Newman heat-flow family, Λ ≥ 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.
Scope: Newman’s separate nonnegativity conjecture is proved.
Proved · version 2
In the classical heat-flow normalization, Λ ≤ 0.2. This is Corollary 2 of the finite-verification paper, applying Polymath15’s criterion with the verified height.
Scope: A published upper bound selected here as a reviewed baseline; newer proposed improvements are separately attributed below.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Bound · Proved
RH is equivalent to Λ ≤ 0. The proved lower bound makes RH equivalent to Λ = 0; it does not prove that upper bound.
Scope: Rodgers and Tao proved Newman’s nonnegativity conjecture: if RH is true, Λ must be exactly zero.
Limit: Λ ≥ 0 alone allows positive Λ, which would make RH false. The required upper bound Λ ≤ 0 remains open.
A distinct scope, not an additive fraction of the conjecture solved.
The two proved inequalities enclose the same normalized constant in [0, 0.2].
An interval containing zero is not a proof that the constant equals zero.
Proved · version 2
For the classical de Bruijn–Newman heat-flow family, Λ ≥ 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.
Scope: Newman’s separate nonnegativity conjecture is proved.
Proved · version 2
In the classical heat-flow normalization, Λ ≤ 0.2. This is Corollary 2 of the finite-verification paper, applying Polymath15’s criterion with the verified height.
Scope: A published upper bound selected here as a reviewed baseline; newer proposed improvements are separately attributed below.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
The de Bruijn–Newman constant Λ is nonnegative.
paraphrase · No upstream full text or code is republished by this record.
Brad Rodgers · Terence Tao · Published
Published in Forum of Mathematics, Pi (2020). The pinned 2021 author revision records a correction to Section 8; earlier manuscript history is not silently substituted.
Source revision: arxiv-v5. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Jul 3, 2021.
The finite-height verification supplies the published criterion giving Λ ≤ 0.2.
paraphrase · No upstream full text or code is republished by this record.
Dave Platt · Tim Trudgian · Published
Published in Bulletin of the London Mathematical Society, DOI 10.1112/blms.12460. Statements here use the pinned author manuscript, Theorem 1 and Corollary 2.
Source revision: arxiv-v1. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Apr 21, 2020.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Source dated Jul 3, 2021 · Observed Oct 10, 2026 · Source passage →