Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Theorem · version 2
Dirichlet GRH implies classical RH
Statement
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
Assumptions
- The trivial conductor-one L-function is ζ(s).
Scope
A one-way specialization; a proof of classical RH supplies no proof for every other character.
Evidence
Source reports
Connections
In this problem · publication v12 · Selected context
Riemann hypothesis →
Conditional implication · Proved
The conductor-one case is classical zeta; this specialization has no asserted converse.
One-way implication
Dirichlet GRH includes classical ζ as the conductor-one case.
The direction is GRH → RH. No converse is asserted.
Details & sources
Generalized RH for Dirichlet L-functions →
Open · version 2
For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.
- Primitive Dirichlet characters of every conductor.
- Zeros in the critical strip; exclude the pole of the principal L-function at s = 1.
Scope: The Dirichlet version of GRH, a broader family than classical ζ.
classical RH holds →
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
- Classical Riemann zeta function; nontrivial zeros in 0 < Re(s) < 1.
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Dirichlet GRH implies classical RH →
Relationship statement · Proved · version 2
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
- The trivial conductor-one L-function is ζ(s).
Scope: A one-way specialization; a proof of classical RH supplies no proof for every other character.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Riemann hypothesis →
Conditional implication · Proved
The conductor-one case is classical zeta; this specialization has no asserted converse.
One-way implication
Dirichlet GRH includes classical ζ as the conductor-one case.
The direction is GRH → RH. No converse is asserted.
Details & sources
Generalized RH for Dirichlet L-functions →
Open · version 2
For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.
- Primitive Dirichlet characters of every conductor.
- Zeros in the critical strip; exclude the pole of the principal L-function at s = 1.
Scope: The Dirichlet version of GRH, a broader family than classical ζ.
classical RH holds →
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
- Classical Riemann zeta function; nontrivial zeros in 0 < Re(s) < 1.
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Dirichlet GRH implies classical RH →
Relationship statement · Proved · version 2
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
- The trivial conductor-one L-function is ζ(s).
Scope: A one-way specialization; a proof of classical RH supplies no proof for every other character.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Alpöge–Furman · simple zeros on the line →
- Theorem B and §1: primitive Dirichlet L-functions; RH not assumed in the proportion bound → · Original passage ↗
Recorded wording
The corresponding all-zeros-on-line conjecture for Dirichlet L-functions is a broader target.
paraphrase · No upstream full text or code is republished by this record.
Levent Alpöge · Ralph Furman · Preprint
August 2026 preprint. Named mathematical authors explicitly state that they verified the proof and take responsibility. Journal acceptance and independent OpenMath expert review are not established.
Revision & attribution
Source revision: arxiv-v2. Observed Oct 10, 2026.
The paper credits Claude (Anthropic) with discovery and writing; Alpöge and Furman with verification, Jarred Sumner with guiding the investigation, and Eric Easley with orchestrating Lean formalization. Company attribution is source-reported, not independently reproduced here.
Source updated Aug 19, 2026.
Bombieri · official RH description →
- §I, PDF page 1 → · Original passage ↗
Recorded wording
Do all nontrivial zeros of the Riemann zeta function have real part 1/2?
paraphrase · No upstream full text or code is republished by this record.
Enrico Bombieri · Published
Official Clay exposition. Historical results and approaches must not be read as current record bounds.
Revision & attribution
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-rh-dirichlet-grh-implies-classical
- Statement version
- 2
- SHA-256
- 9878fd9650265f6e5ab9ca5df00c34fa12c7980bdc74c91cd01cad71e3c9fc32