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Patterson's conjecture on the first moment of cubic Gauss sums over primes

Theorem 1.1: unconditionally, $\sum_{N\pi\le X,\ \pi\equiv1(3)\text{ prime}}G(\pi)=\frac65c_*X^{5/6}/\log X+o(X^{5/6}/\log X)$ , over all primary Eisenstein primes; equivalently $\sum_{p\le X}S_p/(2\sqrt p)\sim\frac35c_*X^{5/6}/\log X$ , matching Wan's Conjecture 4.13. Also cancellation in every fixed nonzero angular mode and for fixed powers $G(\pi)^k$ with $3\nmid k$ , $|k|>1$ . The error term is only little-oh, with no rate; powers $k$ divisible by 3 (part of Patterson's complementary conjecture) remain open. Uses the unconditional Dunn-Radziwill Voronoi formula, not their GRH estimates.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Patterson's conjecture on the first moment of cubic Gauss sums over primes · Patterson's cubic Gauss sum conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: Patterson's conjecture on the first moment of cubic Gauss sums over primes

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No independent mathematician has checked this yet. Checked here: abstract, Theorem 1.1, the history and the normalization section of the TeX source, read against Patterson's conjecture as stated by Dunn-Radziwill and Wan. Lean: Comparator challenge PattersonFirstMoment, declarations OAI.CubicFirstMoment.patterson_firstMoment, patterson_angularComparison and patterson_angularCancellation. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: with Gauss sums at primary primes of $\mathbb Z[\omega]$ built from the cubic residue symbol and $e(\mathrm{Tr}(v/p))$ , the sharp-cutoff sum over primary primes of norm at most $X$ minus $\frac65c_*X^{5/6}/\log X$ is $o(X^{5/6}/\log X)$ ; also the angular statements. That is the headline. Not rebuilt here. The manuscript notes that Dunn-Radziwill print the same constant in two inequivalent normalizations and identifies which one it proves.

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Patterson's conjecture on the first moment of cubic Gauss sums over primes

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Lean Comparator challenge PattersonFirstMoment (OpenAI math release)

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Lean Comparator challenge PattersonFirstMoment (OpenAI math release)

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VibeMathed record: Patterson's conjecture on the first moment of cubic Gauss sums over primes evidence for this event

Lean Comparator challenge PattersonFirstMoment (OpenAI math release) evidence for this event

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