Proved
Scoped theorem in cited source ↗
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
Theorem · version 2
Partial bounds for a fixed Dirichlet character
Statement
For each fixed primitive χ, Theorem A’s lower bounds k₀ for simple on-line zeros and (1+k₀)/2 for distinct zeros hold for L(s,χ), relative to its nontrivial-zero multiplicity count as T → ∞.
Assumptions
- Fix a primitive Dirichlet character χ before taking T → ∞.
- No GRH assumption; the threshold and error constants may depend on the conductor.
Scope
A theorem about a related family, with fixed-character quantifiers.
Evidence
Source reports
Connections
In this problem · publication v9 · Selected context
Riemann hypothesis →
Different statement · related variant · Proved
This extends the method to a GRH variant. It neither proves all zeros lie on the line nor asserts uniform bounds while the conductor grows with T.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Riemann hypothesis →
Different statement · related variant · Proved
This extends the method to a GRH variant. It neither proves all zeros lie on the line nor asserts uniform bounds while the conductor grows with T.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Alpöge–Furman · simple zeros on the line →
- Theorem B, p. 2: fixed primitive χ; constants may depend on conductor → · Original passage ↗
Recorded wording
The simple-on-line and distinct-zero bounds extend to each fixed primitive Dirichlet L-function.
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.
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
Scoped theorem in cited source ↗
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 →
Record details
- Claim ID
- claim-rh-dirichlet-fixed-character-proportions
- Statement version
- 2
- SHA-256
- 5c7fb6a47b93d4294db04d66e1fd9b9c9cfd98dfd3f85d7be9d4d9e6b9aed6ca