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

For a rational prime p≡1(mod3)p\equiv1\pmod 3, Kummer studied the cubic sum Sp=∑x mod pe(x3/p)S_p=\sum_{x\bmod p}e(x^3/p) and observed a bias toward positive values. Heath-Brown and Patterson (1979) proved that the normalized sums are equidistributed on the circle, so any bias lives below the prime-counting scale. Patterson (1978) predicted a bias of exact order X5/6/log⁡XX^{5/6}/\log X; Heath-Brown's cubic large sieve gave O(X5/6+ϵ)O(X^{5/6+\epsilon}) and Dunn-Radziwill (2024) proved a smoothed asymptotic under GRH. With G(π)G(\pi) the normalized cubic Gauss sum at a primary Eisenstein prime and c∗=(2π)2/3/(3Γ(2/3))c_*=(2\pi)^{2/3}/(3\Gamma(2/3)), is ∑Nπ≤XG(π)∼65c∗X5/6/log⁡X\sum_{N\pi\le X}G(\pi)\sim\frac65c_*X^{5/6}/\log X unconditionally?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Analytic number theory; cubic Gauss sums and metaplectic forms
Posed by
S. J. Patterson (1978); first-moment normalization as stated by Dunn and Radziwill (2024, display after Equation (1.8)) and Wan (2021, Conjecture 4.13)
Year posed
1978
Years open
48y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: unconditionally, ∑Nπ≤X, π≡1(3) primeG(π)=65c∗X5/6/log⁡X+o(X5/6/log⁡X)\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 ∑p≤XSp/(2p)∼35c∗X5/6/log⁡X\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(π)kG(\pi)^k with 3∤k3\nmid k, ∣k∣>1|k|>1. The error term is only little-oh, with no rate; powers kk divisible by 3 (part of Patterson's complementary conjecture) remain open. Uses the unconditional Dunn-Radziwill Voronoi formula, not their GRH estimates.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge PattersonFirstMoment).

Verification

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 Z[ω]\mathbb Z[\omega] built from the cubic residue symbol and e(Tr(v/p))e(\mathrm{Tr}(v/p)), the sharp-cutoff sum over primary primes of norm at most XX minus 65c∗X5/6/log⁡X\frac65c_*X^{5/6}/\log X is o(X5/6/log⁡X)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.

Sources

Changelog1 change

Discussion