§I, PDF page 1
Do all nontrivial zeros of the Riemann zeta function have real part 1/2?
Passage details
Passage ID: riemann-hypothesis:rh-classical-target:v1
Source item: §I, PDF page 1
Source record: clay-bombieri-riemann@official-description
Bombieri · official RH description
Published
Open source ↗Overview
Bombieri · official RH description
Official Clay exposition. Historical results and approaches must not be read as current record bounds.
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Passage index · 6
§IV, pp. 6–8: curve zeta functions and Weil’s theorem
The finite-field curve analogue has been proved.
Passage details
Passage ID: riemann-hypothesis:rh-finite-field-curves:v1
Source item: clay-bombieri-riemann#rh-finite-field-curves
Source record: clay-bombieri-riemann@official-description
§I, PDF page 1: Riemann’s 1859 memoir
Riemann’s 1859 memoir relates prime counting to zeta zeros and proposes the all-real-zero assertion for ξ.
Passage details
Passage ID: riemann-hypothesis:rh-origin-history:v1
Source item: §I, PDF page 1: Riemann’s 1859 memoir
Source record: clay-bombieri-riemann@official-description
§II, p. 4, equation π(x) = Li(x) + O(√x log x)
RH is equivalent to the square-root-scale error term for counting primes.
Passage details
Passage ID: riemann-hypothesis:rh-prime-error:v1
Source item: clay-bombieri-riemann#rh-prime-error
Source record: clay-bombieri-riemann@official-description
§III: Hardy, Selberg, Levinson and Conrey; compare current AF Theorem A introduction
Critical-line knowledge developed from infinitely many zeros to positive proportions and sharper quantitative bounds.
Passage details
Passage ID: riemann-hypothesis:rh-proportion-history:v1
Source item: clay-bombieri-riemann#rh-proportion-history
Source record: clay-bombieri-riemann@official-description
§IV finite-field geometry and §V approaches
Geometric, positivity and spectral approaches motivate research but do not supply a classical RH proof.
Passage details
Passage ID: riemann-hypothesis:rh-research-directions:v1
Source item: clay-bombieri-riemann#rh-research-directions
Source record: clay-bombieri-riemann@official-description
Used in
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Riemann hypothesis →
Problem publication v13
classical RH fails →
Source assessment
Open
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
classical RH holds →
Source assessment
Open
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
Proved analogue: curves over finite fields →
Source assessment
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.
Observed Oct 10, 2026
RH is equivalent to the prime-counting error target →
Source assessment
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
Earlier source assessment
Source assessment · superseded assessment
Open
Equivalent assertion; classical RH remains 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.
Observed Oct 10, 2026
Connection
The established equivalence joins two open formulations of the same classical question.
the square-root prime-counting error holds →
Source assessment
Open
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
Problem context · 13 citations
Question
Riemann hypothesis
Further reading
Read the setting and notation before comparing the results.
Connection
The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
History
Riemann connects zeros to prime distribution
Research activity
From infinitely many on-line zeros to proportion bounds
Hardy proved infinitely many zeros lie on the line; later work established positive proportions. The August 2026 paper identifies the previous simple-on-line benchmark as 5/12 and improves it to k₀ ≈ 0.6725007. Its proportion metric differs from finite-height verification.
A historical source’s old numerical records are not today’s best bounds. The selected papers are milestones, not an exhaustive priority history; new submissions stay separately attributed.
History
From infinitely many zeros to positive proportions
Research activity
Geometry, positivity and spectral research directions
The official exposition connects finite-field geometry, positivity criteria and spectral ideas to the classical problem. These explain possible routes and why the finite-field analogue is informative.
An analogy, positivity reformulation or proposed spectral operator does not establish the missing classical theorem. No solution is asserted by this research-direction entry.
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Official description PDF ↗
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Resource revision: official-description
Credits
Recorded authorship
Enrico Bombieri
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- official-description · selectedObserved Oct 10, 2026 · used in current publications
Bombieri · official RH description
Record details
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- Bombieri · official RH description
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- Oct 10, 2026
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