Open
Simple-zero conjecture remains unproved ↗
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 Sep 8, 2026 · Observed Oct 10, 2026
Conjecture · version 2
Separate conjecture: all nontrivial zeros are simple
Statement
Every nontrivial zero of ζ(s) has multiplicity one.
Assumptions
- Classical Riemann zeta function.
- Every nontrivial zero, at every height.
Scope
A multiplicity assertion, separate from the horizontal-position assertion of RH.
Evidence
Source reports
Connections
In this problem · publication v6 · Selected context
Riemann hypothesis →
Different statement · related variant · Open
RH by itself allows repeated on-line zeros. Simple off-line zeros would satisfy this conjecture but violate RH.
Related statements
Line membership and multiplicity are distinct questions. A zero can be on the line and repeated.
The simple-and-on-line proportion theorem addresses an intersection, not all-zero simplicity.
Details & sources
classical RH holds →
Open · version 2
Every nontrivial zero ρ of the meromorphic continuation of ζ(s) has Re(ρ) = 1/2.
- 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.
Separate conjecture: all nontrivial zeros are simple →
Open · version 2
Every nontrivial zero of ζ(s) has multiplicity one.
- Classical Riemann zeta function.
- Every nontrivial zero, at every height.
Scope: A multiplicity assertion, separate from the horizontal-position assertion of RH.
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 →
Different statement · related variant · Open
RH by itself allows repeated on-line zeros. Simple off-line zeros would satisfy this conjecture but violate RH.
Related statements
Line membership and multiplicity are distinct questions. A zero can be on the line and repeated.
The simple-and-on-line proportion theorem addresses an intersection, not all-zero simplicity.
Details & sources
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.
Separate conjecture: all nontrivial zeros are simple →
Open · version 2
Every nontrivial zero of ζ(s) has multiplicity one.
- Classical Riemann zeta function.
- Every nontrivial zero, at every height.
Scope: A multiplicity assertion, separate from the horizontal-position assertion of RH.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Lamzouri · another proof and further proportions →
- §1 discussion of simplicity and critical-line conjectures → · Original passage ↗
Recorded wording
Simplicity of every nontrivial zeta zero is a separate conjecture.
paraphrase · No upstream full text or code is republished by this record.
- Theorem 1.1; definitions of N, N₀ˢ in §1 → · Original passage ↗
Recorded wording
Lamzouri gives another proof of the same k₀ bound.
paraphrase · No upstream full text or code is republished by this record.
Youness Lamzouri · Preprint
Author preprint at the cited revision; journal acceptance has not been established by this curation.
Revision & attribution
Source revision: arxiv-v2. Observed Oct 10, 2026.
Lamzouri gives a human-authored proof inspired by the Claude result. Appendix A reports AxiomProver (Axiom Math) autoformalization; this is a different role from discovering the original bound.
Source updated Sep 8, 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.
Open
Simple-zero conjecture remains unproved ↗
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 Sep 8, 2026 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-rh-simple-zero-variant
- Statement version
- 2
- SHA-256
- 938e1aaaf8d610e6a7f8f2f0b2f6dc0ded1e0b0eaa8f378783fefdfc6e6e82ea