{"paper":{"title":"Asymmetry for the Riemann Hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Walid Oukil","submitted_at":"2025-11-17T15:36:07Z","abstract_excerpt":"In this manuscript, we show that the Riemann zeta function satisfies $\\big(\\zeta(s),\\zeta(1-\\overline{s})\\big)\\neq(0,0)$ for any $s$ in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific \"Rotation Number Hypothesis\"."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that the Riemann zeta function satisfies (ζ(s),ζ(1-¯s))≠(0,0) for any s in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific 'Rotation Number Hypothesis'.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Rotation Number Hypothesis holds for the function that replaces the fractional part function; without an independent justification or verification of this hypothesis, the asymmetry claim does not follow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Riemann zeta function satisfies (ζ(s), ζ(1-¯s)) ≠ (0,0) off the critical line, even under a Rotation Number Hypothesis replacing the fractional part.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7dcf87048ce4cb0bd9ac0a1de3273505df66987df4f1a5e3b25f0dd7a7e556e2"},"source":{"id":"2511.13496","kind":"arxiv","version":12},"verdict":{"id":"3796396a-321c-491b-83e0-2454711e77be","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T20:57:46.663358Z","strongest_claim":"we show that the Riemann zeta function satisfies (ζ(s),ζ(1-¯s))≠(0,0) for any s in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific 'Rotation Number Hypothesis'.","one_line_summary":"The Riemann zeta function satisfies (ζ(s), ζ(1-¯s)) ≠ (0,0) off the critical line, even under a Rotation Number Hypothesis replacing the fractional part.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Rotation Number Hypothesis holds for the function that replaces the fractional part function; without an independent justification or verification of this hypothesis, the asymmetry claim does not follow.","pith_extraction_headline":"The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2511.13496/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"a778a7e739af219378fc703d42346da0d3b933635fc447cfe6bab5b402feebb6"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}