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Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

OpenAI · September 23, 2026

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This is an earlier edition (September 23, 2026). Current edition: Goldfeld's analytic density conjecture and the 2-converse for elliptic curves, October 7, 2026.

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Cite

@misc{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026,
  author = {{OpenAI}},
  title = {{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026}},
  year = {2026}
}

Family 006: Goldfeld’s conjecture: densities and mean analytic rank

Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.

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