{
  "id": "riemann-zeros-proportion",
  "org": "Anthropic",
  "topic": "Number theory",
  "published": "2026-08-10",
  "status": "expert-scrutiny-reported",
  "method": "Mathematicians examined proof; formal artifact reported",
  "math": "Proportion of zeros on Re(s)=½: ≥ 67.2% (claimed)",
  "source": "https://www.anthropic.com/research/riemann-zeta",
  "source_kind": "provider_announcement",
  "checked": "2026-10-29",
  "translations": {
    "en": {
      "title": "More Riemann zeros on the critical line",
      "simple": "Prime numbers follow a mysterious pattern. Claude did not solve the famous Riemann Hypothesis, but its researchers say it raised the minimum known share of certain zeros lying on the crucial line from 41.6% to 67.2%.",
      "research": "An unreleased Claude system improved a lower bound for the proportion of nontrivial zeros of the Riemann zeta function on the critical line. Anthropic reports internal verification, external expert examination and a formal proof artifact.",
      "academic": "For ζ(s), the Riemann Hypothesis asserts that every nontrivial zero has real part 1/2. A positive-proportion theorem gives a lower bound on zeros lying on that line; increasing the proven fraction is distinct from establishing 100%. The announced 67.2% bound and its assumptions must be read with the paper and formalization.",
      "impact": "Advances analytic number theory and showcases AI assistance in delicate estimates, without resolving the distribution of all nontrivial zeros.",
      "caution": "Do not label this a proof of the Riemann Hypothesis. Third-party examination is reported, not independently re-performed by this site."
    },
    "nl": {
      "title": "Meer zèta-nulpunten op de kritieke lijn",
      "simple": "Priemgetallen hebben een raadselachtig patroon. Claude loste de Riemann-hypothese niet op, maar verhoogde volgens Anthropic een bekende ondergrens van 41,6% naar 67,2%.",
      "research": "Het gaat om het aandeel niet-triviale nulpunten van ζ(s) met reëel deel 1/2. Anthropic meldt toetsing door wiskundigen en een formeel bewijs.",
      "academic": "De Riemann-hypothese verlangt 100% van de relevante nulpunten op de kritieke lijn; een verbeterde ondergrens is een ander resultaat.",
      "impact": "Vooruitgang in analytische getaltheorie en bewijsconstructie.",
      "caution": "De Riemann-hypothese blijft onopgelost."
    },
    "de": {
      "title": "Mehr Zeta-Nullstellen auf der kritischen Linie",
      "simple": "Claude löste die Riemannsche Vermutung nicht. Anthropic berichtet aber eine Erhöhung der nachweisbaren Untergrenze von 41,6 % auf 67,2 %.",
      "research": "Untersucht wird der Anteil nichttrivialer Nullstellen von ζ(s) mit Realteil 1/2; Prüfer und ein formales Artefakt werden genannt.",
      "academic": "Die Riemannsche Vermutung verlangt 100 %. Eine höhere bewiesene Untergrenze ist ein anderes mathematisches Ergebnis.",
      "impact": "Fortschritt in analytischer Zahlentheorie und formaler Beweisführung.",
      "caution": "Die Riemannsche Vermutung ist nicht bewiesen."
    },
    "fr": {
      "title": "Davantage de zéros de Riemann sur la droite critique",
      "simple": "Claude n'a pas prouvé l'hypothèse de Riemann. Anthropic annonce cependant une borne inférieure passant de 41,6 % à 67,2 %.",
      "research": "Le résultat concerne la proportion des zéros non triviaux de ζ(s) de partie réelle 1/2 ; des experts ont examiné les travaux.",
      "academic": "L'hypothèse de Riemann exige 100 %. Une borne positive améliorée ne résout pas le problème complet.",
      "impact": "Progrès en théorie analytique des nombres et en preuve formelle.",
      "caution": "L'hypothèse de Riemann demeure ouverte."
    },
    "es": {
      "title": "Más ceros de Riemann en la recta crítica",
      "simple": "Claude no resolvió la hipótesis de Riemann. Anthropic afirma haber elevado un límite inferior del 41,6 % al 67,2 %.",
      "research": "La cota afecta a la proporción de ceros no triviales de ζ(s) con parte real 1/2; se informa de revisión experta.",
      "academic": "La hipótesis de Riemann exige un 100 %. Aumentar un límite inferior demostrado es distinto de resolverla.",
      "impact": "Avance en teoría analítica de números y verificación formal.",
      "caution": "La hipótesis de Riemann sigue abierta."
    },
    "it": {
      "title": "Più zeri di Riemann sulla retta critica",
      "simple": "Claude non ha dimostrato l'ipotesi di Riemann. Anthropic riporta però un miglioramento del limite inferiore dal 41,6% al 67,2%.",
      "research": "Il risultato riguarda gli zeri non banali di ζ(s) con parte reale 1/2; vengono riferite verifiche di esperti.",
      "academic": "L'ipotesi di Riemann richiede il 100%. Una migliore stima inferiore non dimostra l'intera ipotesi.",
      "impact": "Progressi nella teoria analitica dei numeri e nelle prove formali.",
      "caution": "L'ipotesi di Riemann resta aperta."
    }
  },
  "translation_method": "AI-assisted explanatory translations; not native mathematical peer review",
  "url": "https://appliedfrontier.nl/en/discoveries/riemann-zeros-proportion/",
  "as_of": "2026-10-29"
}