{
  "family_id": "006",
  "title": "Goldfeld’s conjecture: densities and mean analytic rank",
  "discipline": "Number theory",
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      "title": "Goldfeld's analytic density conjecture and the 2-converse for elliptic curves",
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    {
      "title": "The mean analytic rank of quadratic twists of elliptic curves",
      "path": "preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf",
      "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf",
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