View source for Patterson's conjecture on cubic Gauss sums

'''Patterson's conjecture on cubic Gauss sums''' predicts that normalized cubic Gauss sums over primes have a small persistent bias: their sum up to ''X'' grows like ''X''<sup>5/6</sup>/log ''X'', although the individual sums are equidistributed on the unit circle. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 25 September 2026, claims an unconditional proof of the first-moment asymptotic. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** For a rational prime ''p'' ≡ 1 mod 3, Kummer's sum ''S''<sub>''p''</sub> is the sum of e<sup>2π''ix''<sup>3</sup>/''p''</sup> over ''x'' mod ''p''. Kummer's numerical observations in 1846 suggested a bias toward positive values. Heath-Brown and Patterson proved in 1979 that the normalized sums are equidistributed on the unit circle, which is compatible with a smaller persistent first moment.
** Patterson predicted a bias on the scale ''X''<sup>5/6</sup>/log ''X''. Heath-Brown's cubic large sieve gave the unconditional upper bound ''O''(''X''<sup>5/6+ε</sup>).
** Dunn and Radziwiłł extended Patterson's metaplectic calculations to a level-aspect Voronoi formula and proved a smoothed first-moment asymptotic under the Generalized Riemann Hypothesis for Hecke ''L''-functions. Wan stated the rational-prime formulation as a conjecture.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''An unconditional first moment for cubic Gauss sums'' states that the sum of the normalized cubic Gauss sums ''G''(π) over all primary primes π of the Eisenstein integers with norm at most ''X'' equals (6/5)''c''<sub>*</sub>''X''<sup>5/6</sup>/log ''X'' + ''o''(''X''<sup>5/6</sup>/log ''X''), where ''c''<sub>*</sub> = (2π)<sup>2/3</sup>/(3Γ(2/3)). Both conjugate primes above a split rational prime are counted, and ''G''(π) + ''G''(π̄) = ''S''<sub>''p''</sub>/√''p''. The manuscript states that this resolves Patterson's first-moment conjecture in the explicit normalizations of Dunn–Radziwiłł and Wan, and it discusses the factor of two between the primary-prime and rational-prime conventions.
* '''Further claims.''' For every fixed integer ℓ, the manuscript compares the sum twisted by (π/|π|)<sup>ℓ</sup> with an elementary model, giving cancellation at the first-moment scale for every fixed nonzero ℓ; the dependence on ℓ is not uniform. It deduces cancellation for the sums of ''G''(π)<sup>''k''</sup> for fixed ''k'' not divisible by 3, part of Patterson's complementary conjecture; nonzero multiples of three are not covered. The error term has no explicit rate.
* ''Method, as described:'' the unconditional level-aspect Voronoi formula of Dunn and Radziwiłł is used, but not their estimates that depend on the Generalized Riemann Hypothesis. The new cancellation is confined to structured prime polynomials, through moments at several lengths and a Gram-matrix estimate, an exact bin-based stopping rule for the prime decomposition, and localization in both norm variables to remove the smooth weight. The manuscript states that it does not improve the cubic large sieve for arbitrary coefficients.
* '''Source''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026 An unconditional first moment for cubic Gauss sums], dated 25 September 2026.
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/lean/docs/023.md lean/docs/023.md] states that the formalization proves the sharp-cutoff asymptotic with the main term above, the angular comparison for every integer mode and cancellation for each fixed nonzero mode, for all primary Eisenstein primes and without an extra conjectural premise. It links the comparator statement PattersonFirstMoment.lean.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 28 October 2026)
|-
| 2026-10-06 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/023-cubic-gauss-first-moment ''The first moment of cubic Gauss sums''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains the manuscript's claimed first-moment asymptotic for cubic Gauss sums over primary Eisenstein primes, including the factor of two between the primary-prime and rational-prime normalizations, corrected dispersion, the stopping-bin decomposition of the prime sum and the scope of the fixed angular modes. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-12 || ''AI Proved What?'', episode 023 [https://aiprovedwhat.com/episodes/023-pattersons-first-moment-cubic-gauss/ Still biased?], remade version of 25 October 2026 || Reading (exposition) || Video explainer of the claim that the normalized cubic Gauss sums over primary primes of the Eisenstein integers with norm up to X add up to 6/5 times c* times X to the power 5/6 over log X plus a smaller error, so that Kummer's normalized sums up to X total about 0.50 X to the power 5/6 over log X, together with cancellation after twisting by any fixed nonzero angular power. The episode states that no rate for the error is claimed and that the companion prediction for powers of the Gauss sum is covered only when 3 does not divide the exponent. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, dated September 25, 2026, released October 6, 52 pages; under expert review: not yet; peer reviewed: not yet; checked in Lean: by OpenAI, the site has not run the check. The page also states that following the imports of OpenAI's Lean statement gives about 3,600 files and 306,000 lines with no unfinished proofs in OpenAI's files, that the result is not listed in OpenAI's formalization catalog, and that it found no public response to the claim.
|-
| 2026-10-17 to 2026-10-28 || Six papers by the agent account bramble_trace under the [https://rstation.ai/problems/riemann_hypothesis/papers Riemann hypothesis problem] of rStation, which describes itself as a research community for AI agents: numbers 5, 52, 54, 62, 65 and 69 || Use and remark || The papers prove estimates for cubic Gauss and Möbius–Gauss sums over the Eisenstein integers within the cubic-theta framework of OpenAI's quasi-Riemann hypothesis manuscript and cite this first-moment manuscript; some use its Lemmas 2.2–2.3 and Section 5, and number 54 states that it does not claim the corrected cubic dispersion developed by Dunn and Radziwiłł and by this manuscript as new. None claims a new zero-free region. || Versioned posts on rStation, operated by Dualverse AI; not peer reviewed. Checked at the source 28 October 2026.
|-
| 2026-10-19 || [https://kokunoyumeto.github.io/open-math-courses-public/courses/the-first-moment-of-cubic-gauss-sums/ ''The first moment of cubic Gauss sums''], a course on Open Mathematics Courses ([https://github.com/kokunoyumeto/open-math-courses-public kokunoyumeto/open-math-courses-public]) || Reading (exposition) || Eight lessons: cubic Gauss sums and Patterson's conjecture, large sieves for cubic characters, the level formula and Type I sums, character moments at two exceptional configurations, corrected dispersion, an exact decomposition of the prime sum, bilinear sums and the sharp cutoff, and angular modes with the powers of Gauss sums whose exponent is not divisible by three. || The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]