we-solve (beta)

Math-2791-A1 · revision r1 · on Math-2791

OpenAI claim summary for Math-2791

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 023; 2026-09-25).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Patterson first moment over Eisenstein primes
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Patterson first moment over Eisenstein primes

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 023; 2026-09-25), the asserted result is that averaging cubic Gauss sums with the prescribed normalization over the entire primary Eisenstein-prime set gives Patterson's main term, and every fixed nonzero angular mode contributes a negligible amount.

Mathematical statement of the claimed result

Manuscript 1.
Set $\omega=e^{2\pi i/3}$, $\mathcal O=\mathbb Z[\omega]$, $N(b)=b\bar b$ and $\operatorname{Tr}(z)=z+\bar z$. For a primary element $b\equiv1\pmod3$, let $\chi_b$ be its cubic residue symbol, zero on nonunits, and define \[ G(b)=N(b)^{-1/2}\sum_{v\bmod b}\chi_b(v)e^{2\pi i\operatorname{Tr}(v/b)},\qquad c_*=(2\pi)^{2/3}/(3\Gamma(2/3)). \] The claimed unconditional sharp-cutoff first moment is \[ \sum_{\substack{N\pi\le X\\\pi\equiv1\pmod3\text{ prime}}}G(\pi) =\tfrac65c_*X^{5/6}/\log X+o(X^{5/6}/\log X)\quad(X\to\infty). \] All primary prime elements are included, in particular both conjugate primes above a split rational prime.

Outline of the argument

For manuscript 1 (family 023; 2026-09-25), angular Hecke characters and a Voronoi transformation give Type I estimates. Corrected dispersion and exceptional-moment bounds are joined to an exact prime decomposition, and smoothing is removed only after each fixed angular mode has been controlled.

Formal verification

For manuscript 1 (family 023; 2026-09-25), OpenAI lists Lean statements for this assertion: averaging cubic Gauss sums with the prescribed normalization over the entire primary Eisenstein-prime set gives Patterson's main term, and every fixed nonzero angular mode contributes a negligible amount.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are PattersonFirstMoment.lean.

Their named declarations include OAI.CubicFirstMoment.patterson_angularCancellation, OAI.CubicFirstMoment.patterson_angularComparison, OAI.CubicFirstMoment.patterson_firstMoment.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources