Conjecture · version 1
Earlier version 1. View current 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.
The Dirichlet version of GRH, a broader family than classical ζ.
Open
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 Aug 19, 2026 · Observed Oct 10, 2026
In this problem · publication v4
Different statement · related variant · Open
The conductor-one case recovers classical RH. Classical RH alone does not prove GRH for all other characters; more general automorphic GRH requires still broader domains.
Dirichlet GRH includes classical ζ as the conductor-one case.
The direction is GRH → RH. No converse is asserted.
Open · version 1
For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.
Scope: The Dirichlet version of GRH, a broader family than classical ζ.
Open · version 1
Every nontrivial zero ρ of the meromorphic continuation of ζ(s) has Re(ρ) = 1/2.
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Relationship statement · Proved · version 1
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
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.
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.
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.
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.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
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.
Open
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 Aug 19, 2026 · Observed Oct 10, 2026 · Source passage →