An unconditional first moment for cubic Gauss sums

Follow-up works

• bramble_trace, Structured cubic and sextic means for angular Gauss coefficients (rStation, v3, 17 October 2026). This work on coefficient-specific angular mean-square estimates cites the manuscript as reference [5] and explicitly distinguishes its prime first moment and angular cancellation scale from the estimates developed there. [AI-generated per source]

Formalization

• OpenAI math release, Patterson's first moment for cubic Gauss sums (Lean scope note 023) (lean/docs/023.md, release of 2026-10-06). The release's own scope note says the Lean development proves the sharp-cutoff asymptotic with main term $\frac65c_*X^{5/6}/\log X$, the angular comparison for every integer mode, and $o(X^{5/6}/\log X)$ cancellation for each fixed nonzero mode; the comparator statement is PattersonFirstMoment.lean. [family-level]

Related prior literature mentioned by others

• Alexander Dunn and Maksym Radziwiłł, Bias in cubic Gauss sums: Patterson's conjecture (Annals of Mathematics 200, 2024; arXiv:2109.07463v3, 14 May 2024). The AI Proved What? exposition and Evidence Press commentary identify this GRH-conditional predecessor, whose level-aspect Voronoi formula and corrected dispersion provide context for the manuscript's claimed unconditional result.

Possibly relevant literature (found by the agent, not checked)

• A. Dunn, Metaplectic cusp forms and the large sieve (arXiv:2403.13151; Algebra & Number Theory 19 (2025)). Proves Type I/II estimates over primes for Fourier coefficients of a cubic metaplectic cusp form via Voronoi summation and Heath-Brown's cubic large sieve, bearing on the paper's prime-sum Type I/bilinear strategy and its remark that it does not improve the cubic large sieve.
• Alexandre de Faveri, Optimal large sieve for fixed order characters (arXiv:2610.04045, 2 October 2026). Its improved large-sieve inequalities for fixed-order Hecke characters may be relevant to the manuscript's use of cubic large-sieve bounds in its structured moment and dispersion estimates.
• Alexandre de Faveri, Alexander Dunn and Jeffrey Hoffstein, Non-orthogonality of the cubic and quartic large sieves via Rankin–Selberg (arXiv:2607.07911v2, 13 July 2026). Its unconditional non-orthogonality results arising from Gauss-sum bias provide a comparison for the manuscript's distinction between cancellation for structured prime coefficients and bounds for arbitrary coefficients.
• C. David, A. Dunn, A. Hamieh and H. Lin, Quartic Gauss sums over primes and metaplectic theta functions (arXiv:2306.11875, v5 2026-01-30). Gives the quartic analogue (Type I and Type II estimates over $\mathbb{Q}(i)$, conjectured moment asymptotics); relevant for whether the paper's unconditional cubic argument has a quartic counterpart.
• Z. Y. Ma, Optimal homological vanishing: cancellation of character sums and Patterson's conjecture over $\mathbb{F}_q[t]$ (arXiv:2606.26440, 2026-06-24). Bounds the bias of higher-order Gauss sums over function fields, a function-field analogue of the Patterson conjecture that the paper proves over $\mathbb{Q}(\omega)$.

Digests and expositions

• Latent Seminar, The first moment of cubic Gauss sums (6 October 2026). This exposition explains the factor of two between primary-prime and rational-prime normalizations, corrected dispersion, the stopping-bin decomposition, and the fixed-mode scope, while explicitly disclaiming independent verification of the complete proof. [family-level]
• AI Proved What?, Still biased? Patterson's first moment for cubic Gauss sums (episode 023, October 2026; status checked 19 October 2026). This AI-assisted video and accompanying explanation illustrate the bias numerically, describe the normalization and fixed-power limitations, and distinguish OpenAI's reported Lean coverage from checks performed by the site. [AI-generated per source] [family-level]
• Gradient Damage, Patterson's Conjecture on Cubic Gauss Sums: OpenAI Claims a Proof (16 October 2026). This short animation illustrates the manuscript's claimed first-moment bias with finite Gauss-sum computations and explicitly presents the animation as illustration rather than proof. [family-level]

News and blogs

• Evidence Press, How did a language model prove a quasi-Riemann hypothesis? (10 October 2026). The article discusses the cubic first-moment manuscript as a possible precursor to OpenAI's quasi-Riemann work, explicitly identifying this discovery-path connection as an inference and noting that the later theorem does not use the first-moment result.