Browse › Number theory › 006 Goldfeld’s conjecture: densities and mean analytic rank
The mean analytic rank of quadratic twists of elliptic curves
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Abstract
For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.
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Related prior literature mentioned by others
- Alexander Smith, The Birch and Swinnerton-Dyer conjecture implies Goldfeld’s conjecture (arXiv:2503.17619, 22 March 2025). The AI Proved What? family exposition highlights this prior theorem on the half-densities of full 2-power Selmer coranks zero and one as the statistical input to the claimed analytic-density result used by both manuscripts. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Antoine Comeau-Lapointe, One-level density of the family of twists of an elliptic curve over function fields (arXiv:2012.09947, December 2020). Gives an unconditional upper bound of 3/2 for the average analytic rank of quadratic twists over F_q(t), a function-field counterpart of the conditional average-rank bounds (Heath-Brown, Fiorilli) that the paper compares with Theorem 1.1.
Digests and expositions
- AI Proved What?, Half and half: Goldfeld’s conjecture for elliptic curve twists (episode 006, October 2026; status checked 19 October 2026). This AI-assisted video and transcript explain the claimed 2-converse, the separate large-rank tail estimate needed for the mean, the signed-squarefree counting convention, and the numerical discussion of the density proof. [AI-generated per source] [family-level]
News and blogs
- Azmat (BinaryVerse AI), Goldfeld’s Conjecture: Why OpenAI’s 50–50 Claim Could Change Number Theory (blog post, 10 October 2026). A non-technical exposition of both manuscripts, separating Smith’s Selmer statistics, the proposed interpolation argument for the 2-converse, and the tail estimate required for mean analytic rank; the author reports finding no completed independent verification as of that date. [family-level]
Elsewhere
- Emergent Mind: family 006
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026,
author = {{OpenAI}},
title = {{The mean analytic rank of quadratic twists of elliptic curves}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf}{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: September 23, 2026
This version updates citations to revised manuscripts in this collection; see the previous version.
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.
Other papers in this family: