Result 023, Number theory

Patterson's first moment for cubic Gauss sums

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X5/6/log⁡XX^{5/6}/\log X. Every fixed nonzero prime-angle Fourier mode has smaller order.

Lean formalization Proof

The bigger picture

Why it matters

Cubic Gauss sums encode how cubes behave modulo primes using complex numbers. The manuscript reports that, across a particular family of primes, these sums do not simply cancel: a positive bias survives with a precise growth rate.

What changes?

The claim concerns normalized sums over all primary Eisenstein primes of norm at most X, including both conjugates, and is unconditional. These are primes in the triangular lattice of Eisenstein integers; 'primary' selects representatives, and norm means squared magnitude. The main term is six fifths times c times X to the five-sixths power, divided by log X. Here c is (2 pi) to the two-thirds power divided by the product of 3 and Gamma of two-thirds.

What does that help mathematicians do?

The manuscript also reports that every fixed nonzero angular Fourier mode contributes a smaller-order sum than the main-term scale. Such a mode weights contributions according to a repeating oscillation in the primes' directions in the complex plane. This rules out a contribution of comparable size from any single fixed nonconstant angular oscillation, distinguishing the overall bias from directional effects. It does not assert uniform control when the frequency grows with X.

Are there practical applications?

The immediate value is foundational: the result supplies a precise benchmark for understanding cancellation in arithmetic exponential sums. Its positive constant and growth rate quantify what survives when contributions from this entire prime family are combined. Applications to other problems would require further arguments; the supplied sources describe an asymptotic theorem, not a computational algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

An unconditional first moment for cubic Gauss sums

September 25, 2026 52 pages

We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is (6/5)c∗X5/6/log⁡X(6/5)c_*X^{5/6}/\log X, where c∗=(2π)2/3/(3Γ(2/3))c_*=(2\pi)^{2/3}/(3\Gamma(2/3)). For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026,
  author = {{OpenAI}},
  title = {{An unconditional first moment for cubic Gauss sums}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf}{OAI:An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/023.md.

Patterson's first moment for cubic Gauss sums

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Patterson's first-moment conjecture concerns the average of normalized cubic Gauss sums over primary Eisenstein primes. The formalization proves the sharp-cutoff asymptotic with main term 65c∗X5/6/log⁡X\frac65c_*X^{5/6}/\log X, where c∗=(2π)2/3/(3Γ(2/3))c_*=(2\pi)^{2/3}/(3\Gamma(2/3)). It also proves the corresponding angular comparison for every integer Fourier mode and cancellation of order o(X5/6/log⁡X)o(X^{5/6}/\log X) for each fixed nonzero mode. All primary Eisenstein primes are included, without an extra conjectural premise.

Comparator links

Result Comparator statement
Patterson first moment and angular cancellation PattersonFirstMoment.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.