{"schema_version":"source-reader-v1","record":{"source_id":"clay-bombieri-riemann","versions":[{"work":{"authors":["Enrico Bombieri"],"kind":"official_description","publication":"published","publication_note":"Official Clay exposition. Historical results and approaches must not be read as current record bounds.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; absence of attribution is not evidence of absence.","artifacts":[{"kind":"pdf","title":"Official description PDF","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","version":"official-description","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."}]},"id":"clay-bombieri-riemann@official-description","source_id":"clay-bombieri-riemann","title":"Bombieri · official RH description","owner":"Enrico Bombieri","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","subject_scope":"Bombieri · official RH description","revision":"official-description","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","attribution":"Enrico Bombieri","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90}],"entries":[{"id":"riemann-hypothesis:rh-classical-target:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"§I, PDF page 1","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"Do all nontrivial zeros of the Riemann zeta function have real part 1/2?","wording_kind":"paraphrase","locator":"§I, PDF page 1","supersedes":null},{"id":"riemann-hypothesis:rh-finite-field-curves:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"clay-bombieri-riemann#rh-finite-field-curves","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"The finite-field curve analogue has been proved.","wording_kind":"paraphrase","locator":"§IV, pp. 6–8: curve zeta functions and Weil’s theorem","supersedes":null},{"id":"riemann-hypothesis:rh-origin-history:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"§I, PDF page 1: Riemann’s 1859 memoir","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"Riemann’s 1859 memoir relates prime counting to zeta zeros and proposes the all-real-zero assertion for ξ.","wording_kind":"paraphrase","locator":"§I, PDF page 1: Riemann’s 1859 memoir","supersedes":null},{"id":"riemann-hypothesis:rh-prime-error:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"clay-bombieri-riemann#rh-prime-error","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"RH is equivalent to the square-root-scale error term for counting primes.","wording_kind":"paraphrase","locator":"§II, p. 4, equation π(x) = Li(x) + O(√x log x)","supersedes":null},{"id":"riemann-hypothesis:rh-proportion-history:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"clay-bombieri-riemann#rh-proportion-history","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"Critical-line knowledge developed from infinitely many zeros to positive proportions and sharper quantitative bounds.","wording_kind":"paraphrase","locator":"§III: Hardy, Selberg, Levinson and Conrey; compare current AF Theorem A introduction","supersedes":null},{"id":"riemann-hypothesis:rh-research-directions:v1","source_version_id":"clay-bombieri-riemann@official-description","item_id":"clay-bombieri-riemann#rh-research-directions","url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","wording":"Geometric, positivity and spectral approaches motivate research but do not supply a classical RH proof.","wording_kind":"paraphrase","locator":"§IV finite-field geometry and §V approaches","supersedes":null}],"problems":[{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13}],"selected_version_id":"clay-bombieri-riemann@official-description","available_versions":[{"id":"clay-bombieri-riemann@official-description","revision":"official-description","observed_on":"2026-10-10","title":"Bombieri · official RH description","publication":"published","content_sha256":null,"used_in_current_publications":true}],"usages":[{"id":"source-use-4a00a9d061fe9e8460c8","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Exact scoped statement in cited source","claim":{"id":"claim-rh-fails","version":2,"slug":"rh-fails","title":"classical RH fails"},"fragment":"claim-rh-fails","assessment":{"id":"riemann-hypothesis:rh-fails:report:v1","entry_id":"riemann-hypothesis:rh-classical-target:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-9cfcd4a2eeba5b9d3c0c","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Exact scoped statement in cited source","claim":{"id":"claim-rh-classical-target","version":3,"slug":"rh-classical-target","title":"classical RH holds"},"fragment":"claim-rh-classical-target","assessment":{"id":"riemann-hypothesis:rh-classical-target:report:v1","entry_id":"riemann-hypothesis:rh-classical-target:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-f5494f60c65f146119b5","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"definition","title":"Critical line","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-b4170f94b521317e61f6","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-d4c7a39f79a7e5b02480","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"reading","title":"Read the setting and notation before comparing the results.","claim":null,"fragment":"provenance","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-bb75ccacc5c45d592327","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"statement","title":"classical RH holds","claim":{"id":"claim-rh-classical-target","version":3,"slug":"rh-classical-target","title":"classical RH holds"},"fragment":"claim-rh-classical-target","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-ca0f44983536f160ef5a","entry_id":"riemann-hypothesis:rh-classical-target:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"statement","title":"classical RH fails","claim":{"id":"claim-rh-fails","version":2,"slug":"rh-fails","title":"classical RH fails"},"fragment":"claim-rh-fails","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-85151a9728e145072988","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Scoped theorem in cited source","claim":{"id":"claim-rh-weil-finite-field-analogue","version":2,"slug":"rh-weil-finite-field-analogue","title":"Proved analogue: curves over finite fields"},"fragment":"claim-rh-weil-finite-field-analogue","assessment":{"id":"riemann-hypothesis:rh-finite-field-curves:report:v1","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1","state":"resolved","source_label":"Scoped theorem in cited source","scope":"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_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-740180027221b1eb6493","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"connection","title":"The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.","claim":null,"fragment":"connections","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-f9f9086894e4948878ff","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-df737f0f400ddd91da0c","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"statement","title":"Proved analogue: curves over finite fields","claim":{"id":"claim-rh-weil-finite-field-analogue","version":2,"slug":"rh-weil-finite-field-analogue","title":"Proved analogue: curves over finite fields"},"fragment":"claim-rh-weil-finite-field-analogue","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-124fad69a73207c48046","entry_id":"riemann-hypothesis:rh-origin-history:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"milestone","title":"Riemann connects zeros to prime distribution","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-09e720af37fbc40fc97d","entry_id":"riemann-hypothesis:rh-origin-history:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-1c9d2993da9338d6d644","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Exact scoped statement in cited source","claim":{"id":"claim-rh-prime-counting-equivalence","version":3,"slug":"rh-prime-counting-equivalence","title":"RH is equivalent to the prime-counting error target"},"fragment":"claim-rh-prime-counting-equivalence","assessment":{"id":"riemann-hypothesis:rh-prime-counting-equivalence-theorem:report:v1","entry_id":"riemann-hypothesis:rh-prime-error:v1","state":"resolved","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":"riemann-hypothesis:rh-prime-error:report:v1"},"superseded":false,"formalization":null,"update":null},{"id":"source-use-2353e0e57b346a148468","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Equivalent assertion; classical RH remains open","claim":{"id":"claim-rh-prime-counting-equivalence","version":3,"slug":"rh-prime-counting-equivalence","title":"RH is equivalent to the prime-counting error target"},"fragment":"claim-rh-prime-counting-equivalence","assessment":{"id":"riemann-hypothesis:rh-prime-error:report:v1","entry_id":"riemann-hypothesis:rh-prime-error:v1","state":"reported_open","source_label":"Equivalent assertion; classical RH remains open","scope":"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_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":null},"superseded":true,"formalization":null,"update":null},{"id":"source-use-6bcba23942750ac797cd","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"assessment","title":"Exact scoped statement in cited source","claim":{"id":"claim-rh-prime-error-target","version":2,"slug":"rh-prime-error-target","title":"the square-root prime-counting error holds"},"fragment":"claim-rh-prime-error-target","assessment":{"id":"riemann-hypothesis:rh-prime-error-target:report:v1","entry_id":"riemann-hypothesis:rh-prime-error:v1","state":"reported_open","source_label":"Exact scoped statement in cited source","scope":"Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-da494d1a800319153dae","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"connection","title":"The established equivalence joins two open formulations of the same classical question.","claim":{"id":"claim-rh-prime-counting-equivalence","version":3,"slug":"rh-prime-counting-equivalence","title":"RH is equivalent to the prime-counting error target"},"fragment":"claim-rh-prime-counting-equivalence","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-7f9497494ca64e3ca7a6","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-bb99ddfdddb2061f0bfb","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"statement","title":"RH is equivalent to the prime-counting error target","claim":{"id":"claim-rh-prime-counting-equivalence","version":3,"slug":"rh-prime-counting-equivalence","title":"RH is equivalent to the prime-counting error target"},"fragment":"claim-rh-prime-counting-equivalence","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-cb664e66b371be9e8473","entry_id":"riemann-hypothesis:rh-prime-error:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"statement","title":"the square-root prime-counting error holds","claim":{"id":"claim-rh-prime-error-target","version":2,"slug":"rh-prime-error-target","title":"the square-root prime-counting error holds"},"fragment":"claim-rh-prime-error-target","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-2310ad44a5714aa7dfe0","entry_id":"riemann-hypothesis:rh-proportion-history:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"activity","title":"From infinitely many on-line zeros to proportion bounds","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"id":"activity-rh-history","title":"From infinitely many on-line zeros to proportion bounds","kind":"research_direction","source_entry_ids":["riemann-hypothesis:rh-proportion-history:v1","riemann-hypothesis:rh-simple-line-af:v1"],"summary":"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.","limitation":"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.","related_claim_ids":["claim-rh-simple-critical-line-proportion"]}},{"id":"source-use-3bfd3209ff2e6988f890","entry_id":"riemann-hypothesis:rh-proportion-history:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"milestone","title":"From infinitely many zeros to positive proportions","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-50fc0070a252c8299709","entry_id":"riemann-hypothesis:rh-proportion-history:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-089f0ddabcced4a7aac5","entry_id":"riemann-hypothesis:rh-research-directions:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"activity","title":"Geometry, positivity and spectral research directions","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"id":"activity-rh-directions","title":"Geometry, positivity and spectral research directions","kind":"research_direction","source_entry_ids":["riemann-hypothesis:rh-research-directions:v1"],"summary":"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.","limitation":"An analogy, positivity reformulation or proposed spectral operator does not establish the missing classical theorem. No solution is asserted by this research-direction entry.","related_claim_ids":[]}},{"id":"source-use-eabb1dcbfcc676bd8dd7","entry_id":"riemann-hypothesis:rh-research-directions:v1","problem":{"slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":13},"kind":"question","title":"Riemann hypothesis","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null}],"linked_entries":[],"linked_sources":[]}}