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RH still requires Λ ≤ 0.","claims":[{"id":"claim-rh-newman-nonnegative","version":2}],"source_entry_ids":["riemann-hypothesis:rh-newman-lower:v1"],"independent_review":"pending"},{"id":"rh-milestone-height","date":{"value":"2020-04-21"},"date_basis":"preprint","category":"bound","title":"Certified verification to height 3,000,175,332,800","summary":"Platt–Trudgian uses rigorous arithmetic and counting to cover every zero through the specified ordinate; it also gives Λ ≤ 0.2.","limitation":"Height is not the number of checked zeros; no assertion about every higher zero follows.","claims":[{"id":"claim-rh-finite-height","version":3},{"id":"claim-rh-newman-upper-bound","version":2}],"source_entry_ids":["riemann-hypothesis:rh-finite-height:v1","riemann-hypothesis:rh-newman-upper:v1"],"independent_review":"pending"},{"id":"rh-milestone-density","date":{"value":"2024-05-31"},"date_basis":"preprint","category":"bound","title":"Guth–Maynard zero-density estimate","summary":"The first preprint introduced the improved exception-count estimate; the selected v2 and 2026 journal publication support the displayed Claim.","limitation":"For fixed σ ≥ 3/4 this bounds possible off-line zeros; it does not exclude every exception.","claims":[{"id":"claim-rh-guth-maynard-zero-density","version":2}],"source_entry_ids":["riemann-hypothesis:rh-zero-density:v1"],"independent_review":"pending"},{"id":"rh-milestone-region","date":{"value":"2026-03-23"},"date_basis":"preprint","category":"bound","title":"An explicit logarithmic zero-free region","summary":"Bellotti–Trudgian–Yang excludes zeros near Re(s) = 1 with an explicit constant and t ≥ 3.","limitation":"The shrinking edge region does not cover the whole half-plane to the right of 1/2.","claims":[{"id":"claim-rh-explicit-zero-free-region-2026","version":2}],"source_entry_ids":["riemann-hypothesis:rh-zero-free-region:v1"],"independent_review":"pending"},{"id":"rh-milestone-claude","date":{"value":"2026-08-19"},"date_basis":"preprint","category":"bound","title":"Above 67.25% simple zeros on the critical line","summary":"Alpöge–Furman v2 presents the Claude-discovered argument, with human mathematical responsibility and reported Lean formalization.","limitation":"An asymptotic simple-and-on-line fraction over total multiplicity is not a percentage of RH solved. OpenMath has not executed the Lean code.","claims":[{"id":"claim-rh-simple-critical-line-proportion","version":2}],"source_entry_ids":["riemann-hypothesis:rh-simple-line-af:v1","riemann-hypothesis:rh-anthropic-formal-audit:v1"],"independent_review":"pending"},{"id":"rh-milestone-lamzouri","date":{"value":"2026-09-08"},"date_basis":"preprint","category":"bound","title":"A second proof and separate counting bounds","summary":"Lamzouri v2 proves the same simple-on-line Claim by a different argument and records further union and average bounds.","limitation":"A second proof does not count as a second increment of coverage. Simple OR on-line is not simple AND on-line.","claims":[{"id":"claim-rh-simple-critical-line-proportion","version":2},{"id":"claim-rh-simple-or-critical-line-proportion","version":2}],"source_entry_ids":["riemann-hypothesis:rh-simple-line-lamzouri:v1","riemann-hypothesis:rh-simple-or-line:v1"],"independent_review":"pending"},{"id":"rh-milestone-openai","date":{"value":"2026-10-06"},"date_basis":"announcement","category":"proposal","title":"OpenAI’s proposed seven-eighths half-plane","summary":"The company release links a manuscript and formal evidence for a fixed zero-free half-plane Re(s) > 7/8.","limitation":"Independent acceptance and proof-chain correspondence are unassessed here. Even a valid bound would leave the region between 1/2 and 7/8.","claims":[{"id":"claim-rh-proposed-seven-eighths-half-plane","version":2}],"source_entry_ids":["riemann-hypothesis:rh-openai-seven-eighths:v1"],"independent_review":"pending"},{"id":"rh-milestone-gomila","date":null,"date_basis":"assessment","category":"proposal","title":"An individual numerical Newman-bound candidate","summary":"The pinned audit package proposes Λ ≤ 0.1787854 and attributes human–AI collaboration.","limitation":"The selected package has no independently established result date or accepted recompile here; the Claim remains proposed.","claims":[{"id":"claim-rh-proposed-newman-improvement","version":2}],"source_entry_ids":["riemann-hypothesis:rh-gomila-newman-candidate:v1"],"independent_review":"pending"}],"progress":[{"method":"rigorous_computation","claim_id":"claim-rh-finite-height","aspect":"Finite-height verification","summary":"Every zero in the stated height range is on the line; rigorous counting excludes missed off-line zeros there.","remaining":"RH still quantifies over every height. A complete check below this bound cannot exclude even one higher off-line zero.","overlaps_with":[],"overlap_note":"A distinct scope, not an additive fraction of the conjecture solved.","category":"special_case"},{"claim_id":"claim-rh-simple-critical-line-proportion","aspect":"Asymptotic critical-line proportion","summary":"An unconditional lower bound above 67.25% for simple zeros on the line; two cited proofs support one Claim.","remaining":"This is a lower bound as height tends to infinity, not a finite-window promise or a percentage of RH solved.","overlaps_with":["claim-rh-finite-height"],"overlap_note":"Finite verification and asymptotic proportion address different regimes; their percentages and heights cannot be added.","category":"bound"},{"claim_id":"claim-rh-guth-maynard-zero-density","aspect":"Zero-density estimate","summary":"The Guth–Maynard estimate improves quantitative bounds on possible zeros away from the critical line.","remaining":"An upper bound on the number of zeros is not proof that there are none; the selected estimate leaves the strip just right of 1/2 untreated.","overlaps_with":["claim-rh-simple-critical-line-proportion"],"overlap_note":"This counts zeros in a horizontal-position range, whereas the proportion bound counts simple zeros on the line.","category":"bound"},{"claim_id":"claim-rh-explicit-zero-free-region-2026","aspect":"Explicit zero-free region","summary":"The March 2026 preprint improves a logarithmic exclusion region near the edge of the strip.","remaining":"The boundary approaches 1 as t increases. RH needs the much stronger exclusion of every zero with Re ρ > 1/2.","overlaps_with":["claim-rh-guth-maynard-zero-density"],"overlap_note":"Excluding a region and bounding counts in a larger region are different conclusions; neither is an additive progress score.","category":"bound"},{"claim_id":"claim-rh-newman-nonnegative","aspect":"Heat-flow lower bound","summary":"Rodgers and Tao proved Newman’s nonnegativity conjecture: if RH is true, Λ must be exactly zero.","remaining":"Λ ≥ 0 alone allows positive Λ, which would make RH false. The required upper bound Λ ≤ 0 remains open.","overlaps_with":[],"overlap_note":"A distinct scope, not an additive fraction of the conjecture solved.","category":"bound"},{"claim_id":"claim-rh-newman-upper-bound","aspect":"Heat-flow upper bound","summary":"The finite-height theorem and Polymath15 criterion yield Λ ≤ 0.2; newer proposed bounds remain separate.","remaining":"A positive upper bound leaves Λ = 0 unresolved. This is a cited baseline, not a claim that no later improvement exists.","overlaps_with":["claim-rh-finite-height","claim-rh-newman-nonnegative"],"overlap_note":"The upper bound depends on the same finite verification as the first result; the two heat-flow bounds constrain one constant.","category":"bound"}],"activities":[{"id":"activity-rh-community-record","title":"Individual AI-assisted submission: reported 67.35478% bound","kind":"formalization","source_entry_ids":["riemann-hypothesis:rh-community-october-record:v1"],"summary":"R. Arun Chandru’s submission reports κ = 66812491/99194876 ≈ 0.6735478050. The independent challenge lists kernel verification and an attestation of GPT-6 Astra use. The displayed bound is asymptotic and counts distinct on-line zeros.","limitation":"The score is not automatically a simple-zero bound. Operator kernel checks, statement faithfulness and expert review are separate; this activity creates no accepted Claim, OpenMath Run or reproduction.","related_claim_ids":["claim-rh-simple-critical-line-proportion"]},{"id":"activity-rh-anthropic-formal","title":"Claude discovery and Anthropic’s reported Lean verification","kind":"formalization","source_entry_ids":["riemann-hypothesis:rh-anthropic-formal-audit:v1"],"summary":"The paper’s human authors take responsibility for a Claude-discovered proof. The pinned Lean repository reports proofs for actual zeta, with standard axioms and comparator verification; dependencies and declarations are linked for inspection.","limitation":"A model-written manuscript, an author-reviewed theorem and a kernel-check report are different evidence. OpenMath has not run the proof; no independent reproduction is asserted.","related_claim_ids":["claim-rh-simple-critical-line-proportion"]},{"id":"activity-rh-axiom-formal","title":"Axiom Math formalization: paper certificate and later repository","kind":"formalization","source_entry_ids":["riemann-hypothesis:rh-axiom-formal-audit:v1"],"summary":"The September paper describes a finite-set certificate and a transfer assuming known analytic inputs. The later pinned repository exports all four actual-zeta proportion targets and reports comparator verification; its dependencies include PrimeNumberTheoremAnd.","limitation":"Do not conflate the early conditional transfer certificate with the later full target types. OpenMath has not audited that expanded proof chain or checked manuscript-to-Lean faithfulness.","related_claim_ids":["claim-rh-simple-critical-line-proportion"]},{"id":"activity-rh-new-ai-proposals","title":"Recent AI-related proposals awaiting assessment","kind":"research_direction","source_entry_ids":["riemann-hypothesis:rh-openai-seven-eighths:v1","riemann-hypothesis:rh-gomila-newman-candidate:v1"],"summary":"OpenAI reports a seven-eighths zero-free half-plane with Lean artifacts; Gomila reports an improved positive Newman upper bound with interval certificates. Each proposed Claim has its own scope and pinned source evidence.","limitation":"These are proposals in this publication, excluded from established progress. No cited source establishes the full classical RH here.","related_claim_ids":["claim-rh-proposed-seven-eighths-half-plane"]},{"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":"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":[]}],"curation":{"checked_on":"2026-10-10","selection":"Six cited scopes of established progress, two equivalent formulations, supporting multiplicity results and four related variants. Recent AI/company/individual proposals are separated from accepted progress. Counts and percentages use explicitly stated conventions. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.","search_scope":"Checked Clay/Bombieri, pinned arXiv theorem statements and PDFs (Platt–Trudgian, Alpöge–Furman, Lamzouri, Guth–Maynard, Bellotti–Trudgian–Yang, Rodgers–Tao), official Anthropic and OpenAI releases, Axiom Math proof repository, individual challenge records and the Gomila candidate package. Targeted Google/DeepMind searches did not establish an RH result for inclusion. This is a selected search as of 10 October 2026, not an exhaustive absence claim.","formal_status":"Anthropic and Axiom repositories report Lean proofs with exact zeta counts. OpenAI supplies targets and proof code for its proposed bound, with collection review unchecked. Comparator challenge files contain deliberate proof placeholders and must be paired with solutions. All code is linked at exact commits; OpenMath has not compiled, axiom-audited or independently reproduced it.","ai_status":"The original proportion theorem credits Claude; Lamzouri’s proof is human-authored with AxiomProver formalization. OpenAI’s recent company preprint is proposed here. Individual GPT-6 Astra use and Gomila’s Claude/ChatGPT/Codex collaboration are self-reported, not company endorsements or proof of RH.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"Every nontrivial zero ρ = β + iγ of ζ with 0 < γ ≤ 3,000,175,332,800 has β = 1/2.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Every nontrivial zero "},{"kind":"math","text":"ρ = β + iγ","latex":"\\rho=\\beta+i\\gamma"},{"kind":"text","text":" of "},{"kind":"math","text":"ζ","latex":"\\zeta"},{"kind":"text","text":" with "},{"kind":"math","text":"0 < γ ≤ 3,000,175,332,800","latex":"0<\\gamma\\le3{,}000{,}175{,}332{,}800"},{"kind":"text","text":" has "},{"kind":"math","text":"β = 1/2","latex":"\\beta=\\frac12"},{"kind":"text","text":"."}]},"id":"claim-rh-finite-height","slug":"rh-finite-height","version":3,"kind":"theorem","title":"RH verified to height 3,000,175,332,800","assumptions":["Classical Riemann zeta function.","Positive ordinate 0 < γ ≤ 3,000,175,332,800; conjugation gives the negative counterpart."],"scope":"A rigorously verified finite height range, using ball arithmetic and a count of all zeros in that range.","source_entry_ids":["riemann-hypothesis:rh-finite-height:v1"],"formal_declarations":[]},{"statement":"Let N(T) count nontrivial zeros with 0 < Im ρ ≤ T, with multiplicity, and N₀ˢ(T) count simple zeros on Re ρ = 1/2. Then liminf as T → ∞ of N₀ˢ(T)/N(T) ≥ k₀ = 3/2 − cot(1/√2)/√2 = 0.6725007037… .","id":"claim-rh-simple-critical-line-proportion","slug":"rh-simple-critical-line-proportion","version":2,"kind":"bound","title":"More than 67.25% simple zeros on the line, asymptotically","assumptions":["Classical ζ; no RH assumption.","Asymptotic proportions as T → ∞; the denominator counts multiplicity."],"scope":"For every ε > 0 the proportion is eventually at least k₀ − ε; dyadic (T,2T] and cumulative (0,T] versions are available.","source_entry_ids":["riemann-hypothesis:rh-simple-line-af:v1","riemann-hypothesis:rh-simple-line-lamzouri:v1"],"formal_declarations":[]},{"statement":"For fixed 3/4 ≤ σ < 1, let N(σ,T) count zeta zeros with Re ρ ≥ σ and |Im ρ| ≤ T, with multiplicity. Then N(σ,T) ≤ T^(15(1−σ)/(3+5σ)+o(1)) as T → ∞; combining classical estimates also gives T^(30(1−σ)/13+o(1)).","id":"claim-rh-guth-maynard-zero-density","slug":"rh-guth-maynard-zero-density","version":2,"kind":"bound","title":"Fewer possible zeros to the right of a fixed line","assumptions":["Fix σ in [3/4,1); let T tend to infinity.","Unconditional large-value and analytic estimates; no RH assumption."],"scope":"This selection states the theorem on a conservative subrange of its domain. The o(1) tends to zero with T, for fixed σ.","source_entry_ids":["riemann-hypothesis:rh-zero-density:v1"],"formal_declarations":[]},{"statement":"ζ(σ + it) ≠ 0 whenever t ≥ 3 and σ > 1 − 1/(4.896 log t), with natural logarithm.","id":"claim-rh-explicit-zero-free-region-2026","slug":"rh-explicit-zero-free-region-2026","version":2,"kind":"bound","title":"An explicit zero-free region near Re(s) = 1","assumptions":["Real t ≥ 3.","Strict inequality σ > 1 − 1/(4.896 log t); classical ζ."],"scope":"A height-dependent region near the right edge of the critical strip, not a fixed half-plane down to Re(s) = 1/2.","source_entry_ids":["riemann-hypothesis:rh-zero-free-region:v1"],"formal_declarations":[]},{"statement":"For the classical de Bruijn–Newman heat-flow family, Λ ≥ 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.","id":"claim-rh-newman-nonnegative","slug":"rh-newman-nonnegative","version":2,"kind":"theorem","title":"The Newman constant satisfies Λ ≥ 0","assumptions":["Classical Riemann ξ heat flow and the cited normalization.","No RH assumption."],"scope":"Newman’s separate nonnegativity conjecture is proved.","source_entry_ids":["riemann-hypothesis:rh-newman-lower:v1"],"formal_declarations":[]},{"statement":"In the classical heat-flow normalization, Λ ≤ 0.2. This is Corollary 2 of the finite-verification paper, applying Polymath15’s criterion with the verified height.","id":"claim-rh-newman-upper-bound","slug":"rh-newman-upper-bound","version":2,"kind":"bound","title":"A cited upper bound Λ ≤ 0.2","assumptions":["Use the same classical heat-flow normalization as the lower bound.","The rigorous finite-height verification satisfies the published Polymath15 criterion."],"scope":"A published upper bound selected here as a reviewed baseline; newer proposed improvements are separately attributed below.","source_entry_ids":["riemann-hypothesis:rh-newman-upper:v1"],"formal_declarations":[]},{"statement":"RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.","id":"claim-rh-prime-counting-equivalence","slug":"rh-prime-counting-equivalence","version":3,"kind":"equivalence","title":"RH is equivalent to the prime-counting error target","assumptions":["Classical prime-counting function for rational primes.","The big-O assertion is for all sufficiently large real x, with one fixed implied constant."],"scope":"An equivalent formulation of the same canonical question.","source_entry_ids":["riemann-hypothesis:rh-prime-error:v1"],"formal_declarations":[]},{"statement":"For the classical de Bruijn–Newman constant, RH is equivalent to Λ = 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.","id":"claim-rh-newman-zero-equivalence","slug":"rh-newman-zero-equivalence","version":3,"kind":"equivalence","title":"RH is equivalent to Λ = 0","assumptions":["Classical heat-flow family.","Use the proved Rodgers–Tao inequality Λ ≥ 0."],"scope":"The equivalence is a theorem; whether the constant is zero remains the unresolved target.","source_entry_ids":["riemann-hypothesis:rh-newman-equivalence:v1"],"formal_declarations":[]},{"statement":"With N_d(T) counting distinct nontrivial zeros and N(T) counting total multiplicity in (0,T], liminf N_d(T)/N(T) ≥ (1+k₀)/2 = 0.8362503518…; k₀ = 3/2 − cot(1/√2)/√2.","id":"claim-rh-distinct-zero-proportion","slug":"rh-distinct-zero-proportion","version":2,"kind":"bound","title":"At least 83.625% distinct zeros, asymptotically","assumptions":["Classical ζ; T → ∞.","N_d counts each distinct zero once; N counts multiplicities."],"scope":"A multiplicity bound, regardless of whether the distinct zeros lie on the line.","source_entry_ids":["riemann-hypothesis:rh-distinct-proportion:v1"],"formal_declarations":[]},{"statement":"Let N_{s∪0}(T) count zeros that are simple or on the critical line, with multiplicity in the union. Then liminf N_{s∪0}(T)/N(T) ≥ C₂ = (1+2√2+2k₀)/(3+2√2) = 0.8876200082… .","id":"claim-rh-simple-or-critical-line-proportion","slug":"rh-simple-or-critical-line-proportion","version":2,"kind":"bound","title":"At least 88.762% simple OR on-line zeros, asymptotically","assumptions":["Classical ζ; T → ∞; k₀ = 3/2 − cot(1/√2)/√2.","Union means simple OR on-line (or both); total N includes multiplicity."],"scope":"A union bound that constrains multiple off-line zeros, without excluding simple off-line zeros.","source_entry_ids":["riemann-hypothesis:rh-simple-or-line:v1"],"formal_declarations":[]},{"statement":"Let N_s(T) count simple zeros anywhere in the strip and N₀(T) count on-line zeros with multiplicity. Then liminf (N_s(T)+N₀(T))/(2N(T)) ≥ (1+k₀)/2 = 0.8362503518… .","id":"claim-rh-average-simple-line-proportion","slug":"rh-average-simple-line-proportion","version":2,"kind":"bound","title":"An average bound for simplicity and line membership","assumptions":["Classical ζ; T → ∞.","k₀ = 3/2 − cot(1/√2)/√2; cumulative positive-ordinate counts."],"scope":"This constrains an average of two different counts.","source_entry_ids":["riemann-hypothesis:rh-average-proportion:v1"],"formal_declarations":[]},{"statement":"For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.","id":"claim-rh-dirichlet-generalized-variant","slug":"rh-dirichlet-generalized-variant","version":2,"kind":"conjecture","title":"Generalized RH for Dirichlet L-functions","assumptions":["Primitive Dirichlet characters of every conductor.","Zeros in the critical strip; exclude the pole of the principal L-function at s = 1."],"scope":"The Dirichlet version of GRH, a broader family than classical ζ.","source_entry_ids":["riemann-hypothesis:rh-dirichlet-grh:v1"],"formal_declarations":[]},{"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 → ∞.","id":"claim-rh-dirichlet-fixed-character-proportions","slug":"rh-dirichlet-fixed-character-proportions","version":2,"kind":"theorem","title":"Partial bounds for a fixed Dirichlet character","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.","source_entry_ids":["riemann-hypothesis:rh-dirichlet-fixed-bound:v1"],"formal_declarations":[]},{"statement":"For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.","id":"claim-rh-weil-finite-field-analogue","slug":"rh-weil-finite-field-analogue","version":2,"kind":"theorem","title":"Proved analogue: curves over finite fields","assumptions":["C is a smooth projective geometrically connected curve over a finite field F_q.","Use its own rational zeta function Z_C, not the classical Riemann ζ."],"scope":"Weil’s RH analogue for curves over finite fields.","source_entry_ids":["riemann-hypothesis:rh-finite-field-curves:v1"],"formal_declarations":[]},{"statement":"Every nontrivial zero of ζ(s) has multiplicity one.","id":"claim-rh-simple-zero-variant","slug":"rh-simple-zero-variant","version":2,"kind":"conjecture","title":"Separate conjecture: all nontrivial zeros are simple","assumptions":["Classical Riemann zeta function.","Every nontrivial zero, at every height."],"scope":"A multiplicity assertion, separate from the horizontal-position assertion of RH.","source_entry_ids":["riemann-hypothesis:rh-simple-zero-conjecture:v1"],"formal_declarations":[]},{"statement":"The source claims ζ(s) ≠ 0 for Re(s) > 7/8, and the same region for every Dirichlet L-function, with principal-character poles at s = 1 excluded. It also claims this for finite-order Hecke L-functions over Q(√−3).","id":"claim-rh-proposed-seven-eighths-half-plane","slug":"rh-proposed-seven-eighths-half-plane","version":2,"kind":"bound","title":"Proposed result: a fixed zero-free half-plane Re(s) > 7/8","assumptions":["Classical ζ and Dirichlet characters of every positive modulus.","For principal Dirichlet characters exclude s = 1; ζ’s pole is not a zero.","The source claims an unconditional theorem; OpenMath has not accepted the proof or audited the formal dependencies."],"scope":"A claimed fixed half-plane, stronger than a region shrinking toward 1, but weaker than RH.","source_entry_ids":["riemann-hypothesis:rh-openai-seven-eighths:v1"],"formal_declarations":[]},{"statement":"The candidate claims Λ ≤ 893927/5000000 = 0.1787854 using X = 6000000185827, t₀ = 129/800 and y₀² = 87677/2500000 in the Polymath15 criterion.","id":"claim-rh-proposed-newman-improvement","slug":"rh-proposed-newman-improvement","version":2,"kind":"bound","title":"Proposed result: Λ ≤ 0.1787854","assumptions":["Classical heat-flow normalization.","Uses published finite RH verification below the barrier, the Polymath15 criterion and the candidate interval certificate package.","No global RH assumption is claimed; certificate correctness and theorem-to-code alignment are unassessed here."],"scope":"A source-reported candidate improvement over the selected published 0.2 baseline.","source_entry_ids":["riemann-hypothesis:rh-gomila-newman-candidate:v1"],"formal_declarations":[]},{"statement":"Every nontrivial zero ρ of the meromorphic continuation of ζ(s) has Re(ρ) = 1/2.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Every nontrivial zero "},{"kind":"math","text":"ρ","latex":"\\rho"},{"kind":"text","text":" of the meromorphic continuation of "},{"kind":"math","text":"ζ(s)","latex":"\\zeta(s)"},{"kind":"text","text":" has "},{"kind":"math","text":"Re(ρ) = 1/2","latex":"\\operatorname{Re}(\\rho)=\\frac12"},{"kind":"text","text":"."}]},"id":"claim-rh-classical-target","slug":"rh-classical-target","version":3,"kind":"conjecture","title":"classical RH holds","assumptions":["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.","source_entry_ids":["riemann-hypothesis:rh-classical-target:v1"],"formal_declarations":[]},{"statement":"There exists a nontrivial zero ρ of ζ(s) with Re(ρ) ≠ 1/2.","id":"claim-rh-fails","slug":"rh-fails","version":2,"kind":"conjecture","title":"classical RH fails","assumptions":["The same classical meromorphic zeta function and critical strip."],"scope":"The logical negation of the all-zero classical target, not a failure of a broader GRH variant.","source_entry_ids":["riemann-hypothesis:rh-classical-target:v1"],"formal_declarations":[]},{"statement":"π(x) = Li(x) + O(√x log x) as x → ∞.","id":"claim-rh-prime-error-target","slug":"rh-prime-error-target","version":2,"kind":"conjecture","title":"the square-root prime-counting error holds","assumptions":["π(x) counts rational primes ≤ x; Li is the logarithmic integral.","A fixed implied constant for all sufficiently large real x."],"scope":"An open target equivalent to classical RH, stronger than the prime number theorem.","source_entry_ids":["riemann-hypothesis:rh-prime-error:v1"],"formal_declarations":[]},{"statement":"Λ = 0 for the classical de Bruijn–Newman heat-flow family H_t and Riemann kernel defined in Rodgers–Tao §1.","id":"claim-rh-newman-zero-target","slug":"rh-newman-zero-target","version":2,"kind":"conjecture","title":"the classical Newman constant is zero","assumptions":["Use exactly the classical H_t normalization of the cited paper.","Rodgers–Tao proves Λ ≥ 0."],"scope":"A heat-flow target equivalent to classical RH; positive upper bounds leave this target open.","source_entry_ids":["riemann-hypothesis:rh-newman-equivalence:v1"],"formal_declarations":[]},{"statement":"If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.","id":"claim-rh-dirichlet-grh-implies-classical","slug":"rh-dirichlet-grh-implies-classical","version":2,"kind":"theorem","title":"Dirichlet GRH implies classical RH","assumptions":["The trivial conductor-one L-function is ζ(s)."],"scope":"A one-way specialization; a proof of classical RH supplies no proof for every other character.","source_entry_ids":["riemann-hypothesis:rh-dirichlet-grh:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"Finite-height verification, asymptotic critical-line proportions, zero-density estimates, a zero-free region and heat-flow bounds establish different scopes. New AI-related claims have separate review labels.","report_ids":["riemann-hypothesis:report:0:v2"],"progress_claim_ids":["claim-rh-finite-height","claim-rh-simple-critical-line-proportion","claim-rh-guth-maynard-zero-density","claim-rh-explicit-zero-free-region-2026","claim-rh-newman-nonnegative","claim-rh-newman-upper-bound"],"remaining":"RH requires every nontrivial zero at every height to lie on Re(s) = 1/2. Finite checks, positive proportions, sparse possible exceptions and positive Newman upper bounds do not settle that assertion; newer proposed results do not change the canonical status."},"statement_kind":"conjecture","problem_id":"problem-riemann-hypothesis","slug":"riemann-hypothesis","title":"Riemann hypothesis","question":"Do all nontrivial zeros of the Riemann zeta function have real part 1/2?","question_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Do all nontrivial zeros of the Riemann zeta function have real part "},{"kind":"math","text":"1/2","latex":"\\frac12"},{"kind":"text","text":"?"}]},"scope":"Classical Riemann ζ(s), meromorphically continued from ∑ n^(−s). Nontrivial zeros are those in 0 < Re(s) < 1; the conjecture requires Re(s) = 1/2 for every such zero, at every height. Trivial negative-even zeros and the pole at s = 1 are outside this question. GRH and finite-field analogues have distinct domains.","scope_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Classical Riemann "},{"kind":"math","text":"ζ(s)","latex":"\\zeta(s)"},{"kind":"text","text":", meromorphically continued from "},{"kind":"math","text":"∑ n^(−s)","latex":"\\sum_{n=1}^{\\infty}n^{-s}"},{"kind":"text","text":". Nontrivial zeros are those in "},{"kind":"math","text":"0 < Re(s) < 1","latex":"0<\\operatorname{Re}(s)<1"},{"kind":"text","text":"; the conjecture requires "},{"kind":"math","text":"Re(s) = 1/2","latex":"\\operatorname{Re}(s)=\\frac12"},{"kind":"text","text":" for every such zero, at every height. Trivial negative-even zeros and the pole at "},{"kind":"math","text":"s = 1","latex":"s=1"},{"kind":"text","text":" are outside this question. GRH and finite-field analogues have distinct domains."}]},"aliases":["Riemann hypothesis","RH","Classical Riemann hypothesis"],"fields":["Number theory","Analysis"],"msc":["11M26"],"topics":["Zeta zeros","Scoped progress","Zero-density estimates","Finite verification","AI-assisted mathematics"],"background":"specialist","scale":"major conjecture","depth":"source-linked","source_entry_ids":["riemann-hypothesis:entry:0:v1","riemann-hypothesis:entry:0:v2","riemann-hypothesis:rh-finite-height:v1","riemann-hypothesis:rh-simple-line-af:v1","riemann-hypothesis:rh-simple-line-lamzouri:v1","riemann-hypothesis:rh-zero-density:v1","riemann-hypothesis:rh-zero-free-region:v1","riemann-hypothesis:rh-newman-lower:v1","riemann-hypothesis:rh-newman-upper:v1","riemann-hypothesis:rh-prime-error:v1","riemann-hypothesis:rh-newman-equivalence:v1","riemann-hypothesis:rh-distinct-proportion:v1","riemann-hypothesis:rh-simple-or-line:v1","riemann-hypothesis:rh-average-proportion:v1","riemann-hypothesis:rh-dirichlet-grh:v1","riemann-hypothesis:rh-dirichlet-fixed-bound:v1","riemann-hypothesis:rh-finite-field-curves:v1","riemann-hypothesis:rh-simple-zero-conjecture:v1","riemann-hypothesis:rh-openai-seven-eighths:v1","riemann-hypothesis:rh-gomila-newman-candidate:v1","riemann-hypothesis:rh-community-october-record:v1","riemann-hypothesis:rh-anthropic-formal-audit:v1","riemann-hypothesis:rh-axiom-formal-audit:v1","riemann-hypothesis:rh-proportion-history:v1","riemann-hypothesis:rh-research-directions:v1","riemann-hypothesis:rh-classical-target:v1","riemann-hypothesis:rh-origin-history:v1"],"status_report_ids":["riemann-hypothesis:report:0:v1","riemann-hypothesis:report:0:v2"],"editorial":null,"literature":[{"title":"Clay current RH status","url":"https://www.claymath.org/millennium/riemann-hypothesis/","note":"Clay reports the full classical conjecture unsolved; observed 10 October 2026. This observation is separate from older theorem dates and recent proposed bounds."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"rh-interpretation-all","title":"Every zero, not almost every zero","summary":"RH is a universal statement at unbounded height. A verified initial interval, a large asymptotic on-line proportion and a sparse exception bound answer different restricted questions.","review":"pending","kind":"interpretation","source_entry_ids":["riemann-hypothesis:rh-classical-target:v1","riemann-hypothesis:rh-finite-height:v1","riemann-hypothesis:rh-simple-line-af:v1"],"claims":[{"id":"claim-rh-classical-target","version":3},{"id":"claim-rh-finite-height","version":3},{"id":"claim-rh-simple-critical-line-proportion","version":2}],"limitation":"Even density one could allow isolated off-line zeros. The selected counts do not estimate proximity to a proof."},{"id":"rh-interpretation-counts","title":"Read the numerator and denominator first","summary":"Simple and on-line, distinct, and simple or on-line are different numerators. The selected asymptotic proportion theorems use total zeros counted with multiplicity as denominator.","review":"pending","kind":"interpretation","source_entry_ids":["riemann-hypothesis:rh-distinct-proportion:v1","riemann-hypothesis:rh-simple-or-line:v1"],"claims":[{"id":"claim-rh-distinct-zero-proportion","version":2},{"id":"claim-rh-simple-or-critical-line-proportion","version":2}],"limitation":"The 83.625% distinct and 88.762% union bounds cannot be substituted for the simple-on-line proportion."},{"id":"rh-approach-analytic","title":"Analytic estimates restrict possible exceptions","summary":"Mollifiers and moment estimates give on-line proportions; large-value estimates give zero-density bounds; explicit estimates exclude edge regions.","review":"pending","kind":"approach","source_entry_ids":["riemann-hypothesis:rh-simple-line-lamzouri:v1","riemann-hypothesis:rh-zero-density:v1","riemann-hypothesis:rh-zero-free-region:v1"],"claims":[{"id":"claim-rh-simple-critical-line-proportion","version":2},{"id":"claim-rh-guth-maynard-zero-density","version":2}],"limitation":"These methods currently leave exceptional zeros possible. Different asymptotic and height assumptions cannot be merged into one all-zero guarantee."},{"id":"rh-approach-certificates","title":"Finite certificates and the heat-flow target","summary":"Rigorous zero counting supplies finite-height certificates and feeds bounds on Λ. The heat-flow formulation makes the remaining target exactly zero.","review":"pending","kind":"approach","source_entry_ids":["riemann-hypothesis:rh-finite-height:v1","riemann-hypothesis:rh-newman-equivalence:v1"],"claims":[{"id":"claim-rh-finite-height","version":3},{"id":"claim-rh-newman-zero-equivalence","version":3}],"limitation":"Positive upper bounds remain insufficient. A reported external formalization or numerical audit must be checked against its precise analytic inputs."},{"id":"rh-approach-geometry","title":"Geometric, positivity and spectral ideas","summary":"Bombieri describes finite-field geometry and positivity or spectral approaches as routes toward understanding zero location.","review":"pending","kind":"approach","source_entry_ids":["riemann-hypothesis:rh-research-directions:v1"],"claims":[{"id":"claim-rh-weil-finite-field-analogue","version":2}],"limitation":"A classical proof still needs the relevant object or positivity statement. The finite-field theorem does not transfer automatically."},{"id":"rh-impact-primes","title":"A uniform asymptotic error for prime counts","summary":"A proof of classical RH would establish π(x) = Li(x) + O(√x log x), controlling deviations from the average distribution at all sufficiently large x.","review":"pending","kind":"impact","source_entry_ids":["riemann-hypothesis:rh-prime-error:v1"],"claims":[{"id":"claim-rh-prime-counting-equivalence","version":3}],"scenario":{"id":"claim-rh-classical-target","version":3},"limitation":"This selected equivalence supplies no explicit numerical constant or practical algorithm. Consequences needing GRH have a stronger premise."},{"id":"rh-impact-newman","title":"The heat-flow threshold would be exactly zero","summary":"Together with the proved nonnegativity theorem, classical RH would identify Λ = 0, fixing the real-zero transition in the classical heat flow.","review":"pending","kind":"impact","source_entry_ids":["riemann-hypothesis:rh-newman-equivalence:v1"],"claims":[{"id":"claim-rh-newman-zero-equivalence","version":3}],"scenario":{"id":"claim-rh-classical-target","version":3},"limitation":"The claim depends on the stated kernel and normalization; it supplies no general result for other heat-flow families."},{"id":"rh-impact-failure","title":"The square-root error target would fail","summary":"An off-line nontrivial zero would refute RH and, by the same established equivalence, rule out the stated O(√x log x) prime-counting estimate.","review":"pending","kind":"impact","source_entry_ids":["riemann-hypothesis:rh-prime-error:v1"],"claims":[{"id":"claim-rh-prime-counting-equivalence","version":3}],"scenario":{"id":"claim-rh-fails","version":2},"limitation":"This does not refute the prime number theorem or quantify a particular finite error. No off-line zero is accepted by this curation."}],"coverage_gaps":["Selected theorem-level milestones, not exhaustive literature coverage, a certified latest-record ranking or a percentage of RH solved.","Independent mathematical and relationship review is pending; bibliographic checking does not establish expert certification.","External Lean and numerical verification reports are distinct from OpenMath execution; no new Run or reproduction is attached.","Recent proof claims remain proposed until assessed. AI attribution is source-reported; search coverage cannot establish that no other AI work exists.","Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved."],"checked_on":"2026-10-10"},"sources":[{"id":"clay-millennium@observed-2026-10-01","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-01","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-01","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"id":"clay-millennium@observed-2026-10-10","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-10","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"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},{"work":{"authors":["Dave Platt","Tim Trudgian"],"kind":"research_paper","publication":"published","publication_note":"Published in Bulletin of the London Mathematical Society, DOI 10.1112/blms.12460. Statements here use the pinned author manuscript, Theorem 1 and Corollary 2.","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":"Verification proof PDF · v1","url":"https://arxiv.org/pdf/2004.09765v1","version":"arXiv:2004.09765v1","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."}]},"id":"arxiv-2004.09765@arxiv-v1","source_id":"arxiv-2004.09765","title":"Platt–Trudgian · rigorous finite verification","owner":"Dave Platt; Tim Trudgian","url":"https://arxiv.org/abs/2004.09765v1","subject_scope":"Platt–Trudgian · rigorous finite verification","revision":"arxiv-v1","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":"2020-04-21","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2004.09765v1","attribution":"Dave Platt; Tim Trudgian","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Levent Alpöge","Ralph Furman"],"kind":"research_paper","publication":"preprint","publication_note":"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.","ai_role":"self_reported","ai_note":"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.","artifacts":[{"kind":"pdf","title":"Proof PDF · v2","url":"https://arxiv.org/pdf/2608.13637v2","version":"arXiv:2608.13637v2","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."},{"kind":"formal_code","title":"Anthropic Lean solution · pinned commit","url":"https://github.com/anthropics/formal-math/blob/2bafb8c88f177284a2123b5fefa2ff84e2365eb6/zeta23/Solution.lean","version":"2bafb8c88f177284a2123b5fefa2ff84e2365eb6","assessment":"external_report","note":"Repository reports a Mathlib riemannZeta proof with standard Lean axioms, comparator checks and no RH premise. Challenge.lean contains statement placeholders; Solution.lean supplies the proof chain. OpenMath has not built or audited the dependencies."},{"kind":"project_notes","title":"Anthropic statement and dependency audit","url":"https://github.com/anthropics/formal-math/blob/2bafb8c88f177284a2123b5fefa2ff84e2365eb6/zeta23/README.md","version":"2bafb8c88f177284a2123b5fefa2ff84e2365eb6","assessment":"external_report","note":"Author/operator audit and toolchain are documented at this commit. This is an external checking report, not an OpenMath Run."}]},"id":"arxiv-2608.13637@arxiv-v2","source_id":"arxiv-2608.13637","title":"Alpöge–Furman · simple zeros on the line","owner":"Levent Alpöge; Ralph Furman","url":"https://arxiv.org/abs/2608.13637v2","subject_scope":"Alpöge–Furman · simple zeros on the line","revision":"arxiv-v2","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":"2026-08-19","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2608.13637v2","attribution":"Levent Alpöge; Ralph Furman","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":30},{"work":{"authors":["Youness Lamzouri"],"kind":"research_paper","publication":"preprint","publication_note":"Author preprint at the cited revision; journal acceptance has not been established by this curation.","ai_role":"self_reported","ai_note":"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.","artifacts":[{"kind":"pdf","title":"Alternative proof PDF · v2","url":"https://arxiv.org/pdf/2609.02882v2","version":"arXiv:2609.02882v2","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."},{"kind":"formal_code","title":"Axiom Math Lean solution · pinned commit","url":"https://github.com/AxiomMath/ZetaZerosV2/blob/1d7edeb4e67096f9da1b37cd4b763e8b73886655/Solution/Basic.lean","version":"1d7edeb4e67096f9da1b37cd4b763e8b73886655","assessment":"external_report","note":"Paper Appendix A describes an early certificate assuming known analytic inputs. The later October repository exports the four proportions for actual Mathlib zeta without such premises in the target types, depends on PrimeNumberTheoremAnd, and reports comparator verification. The dependency proof chain has not been audited or executed by OpenMath."},{"kind":"project_notes","title":"Axiom formal challenge and count definitions","url":"https://github.com/AxiomMath/ZetaZerosV2/blob/1d7edeb4e67096f9da1b37cd4b763e8b73886655/Challenge/Basic.lean","version":"1d7edeb4e67096f9da1b37cd4b763e8b73886655","assessment":"external_report","note":"Challenge targets intentionally use sorry; Solution/Basic.lean supplies proofs. Counts distinguish simple, distinct, on-line multiplicity and their union. A target file alone is not a proof."}]},"id":"arxiv-2609.02882@arxiv-v2","source_id":"arxiv-2609.02882","title":"Lamzouri · another proof and further proportions","owner":"Youness Lamzouri","url":"https://arxiv.org/abs/2609.02882v2","subject_scope":"Lamzouri · another proof and further proportions","revision":"arxiv-v2","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":"2026-09-08","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2609.02882v2","attribution":"Youness Lamzouri","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":30},{"work":{"authors":["Larry Guth","James Maynard"],"kind":"research_paper","publication":"published","publication_note":"Published online 1 March 2026 in Annals of Mathematics 203, pp. 623–675, DOI 10.4007/annals.2026.203.2.6. The author manuscript was revised April 2026; the pinned theorem statement is cited here.","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":"Zero-density proof PDF · v2","url":"https://arxiv.org/pdf/2405.20552v2","version":"arXiv:2405.20552v2","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."},{"kind":"paper","title":"Annals publication and acceptance record","url":"https://annals.math.princeton.edu/2026/203-2/p06","version":"Annals 203 (2026), issue 2","assessment":"paper_proof","note":"Publisher confirms publication and dates; this bibliographic record does not certify an independent OpenMath proof review."}]},"id":"arxiv-2405.20552@arxiv-v2","source_id":"arxiv-2405.20552","title":"Guth–Maynard · zero-density estimates","owner":"Larry Guth; James Maynard","url":"https://arxiv.org/abs/2405.20552v2","subject_scope":"Guth–Maynard · zero-density estimates","revision":"arxiv-v2","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":"2026-04-07","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2405.20552v2","attribution":"Larry Guth; James Maynard","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Chiara Bellotti","Timothy S. Trudgian","Andrew Yang"],"kind":"research_paper","publication":"preprint","publication_note":"Author preprint at the cited revision; journal acceptance has not been established by this curation.","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":"Zero-free-region proof PDF · v1","url":"https://arxiv.org/pdf/2603.21490v1","version":"arXiv:2603.21490v1","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."}]},"id":"arxiv-2603.21490@arxiv-v1","source_id":"arxiv-2603.21490","title":"Bellotti–Trudgian–Yang · explicit zero-free region","owner":"Chiara Bellotti; Timothy S. Trudgian; Andrew Yang","url":"https://arxiv.org/abs/2603.21490v1","subject_scope":"Bellotti–Trudgian–Yang · explicit zero-free region","revision":"arxiv-v1","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":"2026-03-23","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2603.21490v1","attribution":"Chiara Bellotti; Timothy S. Trudgian; Andrew Yang","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Brad Rodgers","Terence Tao"],"kind":"research_paper","publication":"published","publication_note":"Published in Forum of Mathematics, Pi (2020). The pinned 2021 author revision records a correction to Section 8; earlier manuscript history is not silently substituted.","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":"Corrected Newman proof PDF · v5","url":"https://arxiv.org/pdf/1801.05914v5","version":"arXiv:1801.05914v5","assessment":"paper_proof","note":"External source evidence; no OpenMath execution or independent reproduction is attached."}]},"id":"arxiv-1801.05914@arxiv-v5","source_id":"arxiv-1801.05914","title":"Rodgers–Tao · Newman constant is nonnegative","owner":"Brad Rodgers; Terence Tao","url":"https://arxiv.org/abs/1801.05914v5","subject_scope":"Rodgers–Tao · Newman constant is nonnegative","revision":"arxiv-v5","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":"2021-07-03","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/1801.05914v5","attribution":"Brad Rodgers; Terence Tao","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["OpenAI"],"kind":"research_paper","publication":"preprint","publication_note":"Preprint released in the 6 October 2026 collection. The repository’s formalization metadata marks review status unchecked. Recorded here as a proposed result awaiting assessment; formal checking and mathematical faithfulness are separate.","ai_role":"self_reported","ai_note":"OpenAI attributes this collection to an unreleased internal model. The zero-free-region work is an exception to the usual fixed evaluation procedure; the alternate 11/12 writeup was human-edited for readability. Do not infer a public model version or attribute individual GPT-6 submissions to this company release.","artifacts":[{"kind":"pdf","title":"Proposed seven-eighths proof PDF","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","version":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","assessment":"external_report","note":"Manuscript claims a fixed Re(s) > 7/8 region. 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Exact pin preserves later correction history."}]},"id":"openai-quasi-rh@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","source_id":"openai-quasi-rh","title":"OpenAI · reported seven-eighths half-plane","owner":"OpenAI","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/README.md","subject_scope":"OpenAI · reported seven-eighths half-plane","revision":"git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","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":"2026-09-30","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/README.md","attribution":"OpenAI","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":30},{"work":{"authors":["Jude Gomila"],"kind":"research_project","publication":"project","publication_note":"Public candidate audit package, not yet peer reviewed. The author reports interval certificates; AI referee checks do not establish independent human mathematical acceptance.","ai_role":"self_reported","ai_note":"The author discloses human–AI collaboration using Claude and ChatGPT/Codex. 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Arun Chandru (submitter); Independent riemannzeta.fun project (operator)","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":30}],"entries":[{"id":"riemann-hypothesis:entry:0:v1","source_version_id":"clay-millennium@observed-2026-10-01","item_id":"https://www.claymath.org/millennium/riemann-hypothesis/","url":"https://www.claymath.org/millennium/riemann-hypothesis/","wording":"Do all nontrivial zeros of the Riemann zeta function have real part 1/2?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/riemann-hypothesis/","supersedes":null},{"id":"riemann-hypothesis:entry:0:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"https://www.claymath.org/millennium/riemann-hypothesis/","url":"https://www.claymath.org/millennium/riemann-hypothesis/","wording":"Do all nontrivial zeros of the Riemann zeta function have real part 1/2?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/riemann-hypothesis/","supersedes":"riemann-hypothesis:entry:0:v1"},{"id":"riemann-hypothesis:rh-finite-height:v1","source_version_id":"arxiv-2004.09765@arxiv-v1","item_id":"arxiv-2004.09765#rh-finite-height","url":"https://arxiv.org/abs/2004.09765v1","wording":"All nontrivial zeros with positive imaginary part at most 3,000,175,332,800 lie on the critical line.","wording_kind":"paraphrase","locator":"Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting","supersedes":null},{"id":"riemann-hypothesis:rh-simple-line-af:v1","source_version_id":"arxiv-2608.13637@arxiv-v2","item_id":"arxiv-2608.13637#rh-simple-line-af","url":"https://arxiv.org/abs/2608.13637v2","wording":"An unconditional asymptotic lower bound k₀ = 3/2 − cot(1/√2)/√2 holds for simple critical-line zeros relative to total multiplicity.","wording_kind":"paraphrase","locator":"Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1","supersedes":null},{"id":"riemann-hypothesis:rh-simple-line-lamzouri:v1","source_version_id":"arxiv-2609.02882@arxiv-v2","item_id":"arxiv-2609.02882#rh-simple-line-lamzouri","url":"https://arxiv.org/abs/2609.02882v2","wording":"Lamzouri gives another proof of the same k₀ bound.","wording_kind":"paraphrase","locator":"Theorem 1.1; definitions of N, N₀ˢ in §1","supersedes":null},{"id":"riemann-hypothesis:rh-zero-density:v1","source_version_id":"arxiv-2405.20552@arxiv-v2","item_id":"arxiv-2405.20552#rh-zero-density","url":"https://arxiv.org/abs/2405.20552v2","wording":"A new bound limits the number of zeros to the right of a fixed vertical line.","wording_kind":"paraphrase","locator":"Theorem 1.2 and equation (1.4), p. 2","supersedes":null},{"id":"riemann-hypothesis:rh-zero-free-region:v1","source_version_id":"arxiv-2603.21490@arxiv-v1","item_id":"arxiv-2603.21490#rh-zero-free-region","url":"https://arxiv.org/abs/2603.21490v1","wording":"There are no zeta zeros for t ≥ 3 and σ > 1 − 1/(4.896 log t).","wording_kind":"paraphrase","locator":"Theorem 1, p. 2 (strict inequality in theorem statement)","supersedes":null},{"id":"riemann-hypothesis:rh-newman-lower:v1","source_version_id":"arxiv-1801.05914@arxiv-v5","item_id":"arxiv-1801.05914#rh-newman-lower","url":"https://arxiv.org/abs/1801.05914v5","wording":"The de Bruijn–Newman constant Λ is nonnegative.","wording_kind":"paraphrase","locator":"Theorem 1.1; §1 normalization and RH equivalence","supersedes":null},{"id":"riemann-hypothesis:rh-newman-upper:v1","source_version_id":"arxiv-2004.09765@arxiv-v1","item_id":"arxiv-2004.09765#rh-newman-upper","url":"https://arxiv.org/abs/2004.09765v1","wording":"The finite-height verification supplies the published criterion giving Λ ≤ 0.2.","wording_kind":"paraphrase","locator":"Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1","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-newman-equivalence:v1","source_version_id":"arxiv-1801.05914@arxiv-v5","item_id":"arxiv-1801.05914#rh-newman-equivalence","url":"https://arxiv.org/abs/1801.05914v5","wording":"After the nonnegativity theorem, RH is equivalent to Λ = 0.","wording_kind":"paraphrase","locator":"§1: RH iff Λ ≤ 0; 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RH not assumed in the proportion bound","supersedes":null},{"id":"riemann-hypothesis:rh-dirichlet-fixed-bound:v1","source_version_id":"arxiv-2608.13637@arxiv-v2","item_id":"arxiv-2608.13637#rh-dirichlet-fixed-bound","url":"https://arxiv.org/abs/2608.13637v2","wording":"The simple-on-line and distinct-zero bounds extend to each fixed primitive Dirichlet L-function.","wording_kind":"paraphrase","locator":"Theorem B, p. 2: fixed primitive χ; constants may depend on conductor","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-simple-zero-conjecture:v1","source_version_id":"arxiv-2609.02882@arxiv-v2","item_id":"arxiv-2609.02882#rh-simple-zero-conjecture","url":"https://arxiv.org/abs/2609.02882v2","wording":"Simplicity of every nontrivial zeta zero is a separate conjecture.","wording_kind":"paraphrase","locator":"§1 discussion of simplicity and critical-line conjectures","supersedes":null},{"id":"riemann-hypothesis:rh-openai-seven-eighths:v1","source_version_id":"openai-quasi-rh@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","item_id":"openai-quasi-rh#rh-openai-seven-eighths","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/README.md","wording":"OpenAI claims a fixed zero-free half-plane Re(s) > 7/8 for zeta and all Dirichlet L-functions.","wording_kind":"paraphrase","locator":"Theorem 1.1; formalization scope family 003; comparator OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re","supersedes":null},{"id":"riemann-hypothesis:rh-gomila-newman-candidate:v1","source_version_id":"gomila-newman-candidate@git-a74738deb6d5e0f76887cb36901da08b68dca705","item_id":"gomila-newman-candidate#rh-gomila-newman-candidate","url":"https://github.com/judegomila/dbn-lambda-01787854-candidate-audit/blob/a74738deb6d5e0f76887cb36901da08b68dca705/README.md","wording":"The candidate reports Λ ≤ 893927/5000000 = 0.1787854 by Polymath15 and interval certificates.","wording_kind":"paraphrase","locator":"README exact parameters; release manuscript; OPEN_REVIEW_QUESTIONS","supersedes":null},{"id":"riemann-hypothesis:rh-community-october-record:v1","source_version_id":"riemann-community-record@git-d95a6d7d11d7836306ec7ec1eb5be3478663718e","item_id":"riemann-community-record#rh-community-october-record","url":"https://github.com/josusanmartin/riemann/blob/d95a6d7d11d7836306ec7ec1eb5be3478663718e/submissions/master-of-puppets/submission.json","wording":"An independent submission reports κ = 66812491/99194876 for distinct critical-line zeros over the multiplicity count.","wording_kind":"paraphrase","locator":"Submission master-of-puppets; 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this is not a resolution of classical RH or independent OpenMath expert assessment.","source_date":"2026-03-23","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2603.21490v1","supersedes":null},{"id":"riemann-hypothesis:rh-newman-lower:report:v1","entry_id":"riemann-hypothesis:rh-newman-lower: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":"2021-07-03","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/1801.05914v5","supersedes":null},{"id":"riemann-hypothesis:rh-newman-upper:report:v1","entry_id":"riemann-hypothesis:rh-newman-upper: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":"2020-04-21","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2004.09765v1","supersedes":null},{"id":"riemann-hypothesis:rh-prime-error:report:v1","entry_id":"riemann-hypothesis:rh-prime-error:v1","state":"reported_open","source_label":"Equivalent assertion; 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this is not a resolution of classical RH or independent OpenMath expert assessment.","source_date":"2026-08-19","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2608.13637v2","supersedes":null},{"id":"riemann-hypothesis:rh-dirichlet-fixed-bound:report:v1","entry_id":"riemann-hypothesis:rh-dirichlet-fixed-bound: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":"2026-08-19","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2608.13637v2","supersedes":null},{"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},{"id":"riemann-hypothesis:rh-simple-zero-conjecture:report:v1","entry_id":"riemann-hypothesis:rh-simple-zero-conjecture:v1","state":"reported_open","source_label":"Simple-zero conjecture remains unproved","scope":"Only the exact statement, domain and revision cited by this Claim; 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this is not a resolution of classical RH or independent OpenMath expert assessment.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://github.com/judegomila/dbn-lambda-01787854-candidate-audit/blob/a74738deb6d5e0f76887cb36901da08b68dca705/README.md","supersedes":null},{"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},{"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},{"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},{"id":"riemann-hypothesis:rh-newman-zero-target:report:v1","entry_id":"riemann-hypothesis:rh-newman-equivalence: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://arxiv.org/abs/1801.05914v5","supersedes":null},{"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"},{"id":"riemann-hypothesis:rh-newman-zero-equivalence-theorem:report:v1","entry_id":"riemann-hypothesis:rh-newman-equivalence: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://arxiv.org/abs/1801.05914v5","supersedes":"riemann-hypothesis:rh-newman-equivalence:report:v1"},{"id":"riemann-hypothesis:rh-dirichlet-grh-implies-classical:report:v1","entry_id":"riemann-hypothesis:rh-dirichlet-grh: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://arxiv.org/abs/2608.13637v2","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"Finite-height verification, asymptotic critical-line proportions, zero-density estimates, a zero-free region and heat-flow bounds establish different scopes. 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Finite checks, positive proportions, sparse possible exceptions and positive Newman upper bounds do not settle that assertion; newer proposed results do not change the canonical status.","views":["open","partial"],"knowledge_categories":["Special cases proved","Bounds established"]}},"selected_context":{"assessment_history":[{"report_id":"riemann-hypothesis:rh-finite-field-curves:report:v1","source_id":"clay-bombieri-riemann","source_version_id":"clay-bombieri-riemann@official-description","source_title":"Bombieri · official RH description","source_label":"Scoped theorem in cited source","state":"resolved","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.","observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf","entry_id":"riemann-hypothesis:rh-finite-field-curves:v1"}],"problem_id":"problem-riemann-hypothesis","slug":"riemann-hypothesis","title":"Riemann hypothesis","publication_version":9,"relationship":{"claim":{"id":"claim-rh-weil-finite-field-analogue","version":2},"relation":"related_variant","relationship_note":"This is proved in another mathematical domain. 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Its geometric proof gives insight but does not establish classical RH; it is not counted as a proved special case of ζ.","result":"proved","report_ids":["riemann-hypothesis:rh-finite-field-curves:report:v1"],"independent_review":"pending"},{"statement":"Every nontrivial zero ρ of the meromorphic continuation of ζ(s) has Re(ρ) = 1/2.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Every nontrivial zero "},{"kind":"math","text":"ρ","latex":"\\rho"},{"kind":"text","text":" of the meromorphic continuation of "},{"kind":"math","text":"ζ(s)","latex":"\\zeta(s)"},{"kind":"text","text":" has "},{"kind":"math","text":"Re(ρ) = 1/2","latex":"\\operatorname{Re}(\\rho)=\\frac12"},{"kind":"text","text":"."}]},"id":"claim-rh-classical-target","slug":"rh-classical-target","version":3,"kind":"conjecture","title":"classical RH holds","assumptions":["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.","source_entry_ids":["riemann-hypothesis:rh-classical-target:v1"],"formal_declarations":[],"relation":"primary_formulation","relationship_note":"Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.","result":"unproved","report_ids":["riemann-hypothesis:rh-classical-target:report:v1"],"independent_review":"pending"}],"entries":[{"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-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}],"sources":[{"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}]}}}