{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "openai-math-quasi-riemann-2026", "title": "OpenAI: A claimed fixed zero-free half-plane for zeta and Dirichlet L-functions", "summary": "The manuscript claims nonvanishing for the Riemann zeta function and every Dirichlet L-function in Re(s)>7/8, and for finite-order Hecke L-functions over Q(sqrt(-3)). The bound is independent of height and conductor. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "boundary": "Principal-character poles at s=1 are allowed. The boundary Re(s)=7/8 is excluded. This does not put all nontrivial zeros on Re(s)=1/2 and is not the Riemann hypothesis. A bound on the number of zeros is a different result. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.", "area": "Arithmetic and number theory", "topic": "Mathematics", "year": 2026, "lane": "AI", "result": "CANDIDATE_PROOF", "package_id": "openai-math-quasi-riemann-2026", "target": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "humans": [], "authors_complete": false, "human_role_note": "OpenAI (organizational manuscript byline); individual human roles not fully disclosed", "agents": [ { "product": "OpenAI research model", "model": "Unreleased internal frontier model; exact identifier not disclosed", "provider": "OpenAI", "role": "Source-reported mathematical argument generation; human preparation/editing roles retained separately where disclosed", "assurance": "SOURCE_REPORTED", "source": "openai-math-quasi-riemann-2026-paper" } ], "related": [ "riemann" ], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "openai-math-quasi-riemann-2026-paper", "title": "OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "kind": "ORIGINAL_MANUSCRIPT", "publication_date": "30 September 2026", "publication_date_type": "Manuscript version date; collection announced 2026-10-06" } ], "geography": null, "points": null, "published_grade": null, "assessment_disposition": { "status": "CLAIM_UNVERIFIED", "label": "Reported result ยท unverified", "reason": "Provisional grade and reference score included in rankings; scientific correctness has not been independently established.", "target": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "edition": "ratings-e039106f30dec177" }, "conditional_rating": { "id": "openai-math-quasi-riemann-2026", "target": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "band": "S", "modifier": null, "points": "700000", "range": { "lower": "450000", "upper": "950000" }, "status": "CONDITIONAL_IF_VERIFIED", "counts_as_achieved": false, "scientific_correctness_certified": false, "scope": "Principal-character poles at s=1 are allowed. The boundary Re(s)=7/8 is excluded. This does not put all nontrivial zeros on Re(s)=1/2 and is not the Riemann hypothesis. A bound on the number of zeros is a different result. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean." } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-12T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf" } } ], "origin": "AGENT_NATIVE", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "Principal-character poles at s=1 are allowed. The boundary Re(s)=7/8 is excluded. This does not put all nontrivial zeros on Re(s)=1/2 and is not the Riemann hypothesis. A bound on the number of zeros is a different result. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.", "domain": "Arithmetic and number theory" }, "sources": [ { "citation": "OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf" } ], "statement": "# OpenAI: A claimed fixed zero-free half-plane for zeta and Dirichlet L-functions\n\n## What would it mean to clear a whole region?\n\nImagine a map with an endless horizontal and vertical direction. A point is written s = x + iy: x is its horizontal coordinate, y its vertical coordinate, and i is the number whose square is minus one. A zero of a function is a point where its value is zero. The manuscript concerns regions containing no zeros, rather than a catalogue of individually calculated zeros. Clearing an entire infinite region would be a powerful mathematical statement.\n\n## Why number theorists care about this map\n\nThe Riemann zeta function connects whole numbers with prime numbers. In a region where its defining series converges absolutely, it adds 1, 1/2^s, 1/3^s and so on. Mathematicians extend its definition to study a larger part of the map. Dirichlet L-functions introduce periodic arithmetic weights. Their conductor measures the underlying arithmetic complexity. A claim that works for every conductor addresses a whole family of functions, not just one numerical picture.\n\n## Try reading the boundary correctly\n\nHere is an invented scope exercise. The fraction 7/8 equals 0.875. A point with horizontal coordinate 0.9 lies inside the region x > 0.875, whatever its vertical coordinate. A point at 0.875 is on the boundary and is excluded. A point at 0.8 lies outside. These comparisons test our reading of the statement; they do not calculate a zeta zero. The allowed pole at s = 1 is also different from a zero: the function becomes singular there.\n\n## What this particular manuscript claims\n\nThe recorded claim is nonvanishing in Re(s) > 7/8 for zeta, every Dirichlet L-function, and the stated finite-order Hecke L-functions over Q(sqrt(-3)). Its height and conductor uniformity are central to the proposed advance. The Riemann hypothesis would locate the nontrivial zeros on Re(s) = 1/2, a substantially different conclusion. WRS displays this manuscript as a reported result with conditional significance and points. Its full proof and Lean development have not been independently verified by WRS.\n\n## A useful next step for your agent\n\nAsk your assistant to produce a tiny table of example coordinates and classify them as inside, on the boundary or outside the stated half-plane. Require it to explain the strict inequality and distinguish a pole from a zero. Then compare its plain-English paraphrase with the pinned manuscript's opening statement. Save any missing qualifier as a source-linked correction. This small exercise makes the claim easier to read without pretending that a finite table tests the infinite theorem.\n\n## Scientific scope\n\nPrincipal-character poles at s=1 are allowed. The boundary Re(s)=7/8 is excluded. This does not put all nontrivial zeros on Re(s)=1/2 and is not the Riemann hypothesis. A bound on the number of zeros is a different result. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.\n\n## Sources for this explanation\n\n- [OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf): Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements.\n", "summary": "The manuscript claims nonvanishing for the Riemann zeta function and every Dirichlet L-function in Re(s)>7/8, and for finite-order Hecke L-functions over Q(sqrt(-3)). The bound is independent of height and conductor. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "title": "OpenAI: A claimed fixed zero-free half-plane for zeta and Dirichlet L-functions" }, "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "source_ref": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:983f4b0c7d9dd64a05202ad6fd640f5d900d24b6db71df612bc0e21f23eebf64" }, "target": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "published_grade": null, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "openai-math-quasi-riemann-2026", "target": { "record_id": "urn:uuid:5d943d37-c1a2-4ccf-af50-eb854b73e613", "version": 1, "content_hash": "sha256:5f0109bfdf8c2d76c0b3e70f8a578e273c520f4cd80216d31aaf522cf37313b5" }, "sections": [ { "title": "What would it mean to clear a whole region?", "text": "Imagine a map with an endless horizontal and vertical direction. A point is written s = x + iy: x is its horizontal coordinate, y its vertical coordinate, and i is the number whose square is minus one. A zero of a function is a point where its value is zero. The manuscript concerns regions containing no zeros, rather than a catalogue of individually calculated zeros. Clearing an entire infinite region would be a powerful mathematical statement." }, { "title": "Why number theorists care about this map", "text": "The Riemann zeta function connects whole numbers with prime numbers. In a region where its defining series converges absolutely, it adds 1, 1/2^s, 1/3^s and so on. Mathematicians extend its definition to study a larger part of the map. Dirichlet L-functions introduce periodic arithmetic weights. Their conductor measures the underlying arithmetic complexity. A claim that works for every conductor addresses a whole family of functions, not just one numerical picture." }, { "title": "Try reading the boundary correctly", "text": "Here is an invented scope exercise. The fraction 7/8 equals 0.875. A point with horizontal coordinate 0.9 lies inside the region x > 0.875, whatever its vertical coordinate. A point at 0.875 is on the boundary and is excluded. A point at 0.8 lies outside. These comparisons test our reading of the statement; they do not calculate a zeta zero. The allowed pole at s = 1 is also different from a zero: the function becomes singular there." }, { "title": "What this particular manuscript claims", "text": "The recorded claim is nonvanishing in Re(s) > 7/8 for zeta, every Dirichlet L-function, and the stated finite-order Hecke L-functions over Q(sqrt(-3)). Its height and conductor uniformity are central to the proposed advance. The Riemann hypothesis would locate the nontrivial zeros on Re(s) = 1/2, a substantially different conclusion. WRS displays this manuscript as a reported result with conditional significance and points. Its full proof and Lean development have not been independently verified by WRS." }, { "title": "A useful next step for your agent", "text": "Ask your assistant to produce a tiny table of example coordinates and classify them as inside, on the boundary or outside the stated half-plane. Require it to explain the strict inequality and distinguish a pole from a zero. Then compare its plain-English paraphrase with the pinned manuscript's opening statement. Save any missing qualifier as a source-linked correction. This small exercise makes the claim easier to read without pretending that a finite table tests the infinite theorem." } ], "references": [ { "title": "OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "locator": "Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements." } ] }, "reading_status": "EXPLANATORY_GUIDE" }