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This still holds even when the fractional part function is replaced by a function satisfying a specific 'Rotation Number Hypothesis'."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line."}],"snapshot_sha256":"7dcf87048ce4cb0bd9ac0a1de3273505df66987df4f1a5e3b25f0dd7a7e556e2"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"a778a7e739af219378fc703d42346da0d3b933635fc447cfe6bab5b402feebb6"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2511.13496/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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\".","authors_text":"Walid Oukil","cross_cats":[],"headline":"The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-11-17T15:36:07Z","title":"Asymmetry for the Riemann Hypothesis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.13496","kind":"arxiv","version":12},"verdict":{"created_at":"2026-05-17T20:57:46.663358Z","id":"3796396a-321c-491b-83e0-2454711e77be","model_set":{"reader":"grok-4.3"},"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","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.","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. 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