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The binary corrected Elliott conjecture for ordinary averages

Elliott (1992) conjectured that correlations 1N∑n≤Nf1(n+h1)⋯fk(n+hk)\frac1N\sum_{n\le N}f_1(n+h_1)\cdots f_k(n+h_k) of multiplicative functions bounded by one tend to zero when one factor does not pretend to be a twisted character. Matomaki, Radziwill and Tao showed the original hypothesis is insufficient for complex functions and corrected it: with D(f,χnit;N)2=∑p≤N(1−Re f(p)χ(p)‾p−it)/pD(f,\chi n^{it};N)^2=\sum_{p\le N}(1-\mathrm{Re}\,f(p)\overline{\chi(p)}p^{-it})/p, require inf⁡∣t∣≤ND(f,χnit;N)→∞\inf_{|t|\le N}D(f,\chi n^{it};N)\to\infty for every fixed Dirichlet character χ\chi. Tao proved the logarithmically averaged binary case, and Tao-Teravainen and Klurman-Mangerel-Teravainen proved it along most scales. For multiplicative f1,f2:N→{∣z∣≤1}f_1,f_2:\mathbb{N}\to\{|z|\le1\} with one factor uniformly nonpretentious and fixed distinct h1,h2h_1,h_2, does 1N∑n≤Nf1(n+h1)f2(n+h2)→0\frac1N\sum_{n\le N}f_1(n+h_1)f_2(n+h_2)\to0?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Analytic number theory; pretentious multiplicative functions
Posed by
P. D. T. A. Elliott (Notas Soc. Mat. Chile, 1992); corrected form by Matomaki, Radziwill and Tao (2015)
Year posed
1992
Years open
34y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For multiplicative f1,f2:N→{∣z∣≤1}f_1,f_2:\mathbb{N}\to\{|z|\le1\}, at least one satisfying inf⁡∣t∣≤ND(fj,χnit;N)→∞\inf_{|t|\le N}D(f_j,\chi n^{it};N)\to\infty for every fixed Dirichlet character, and fixed distinct h1,h2≥0h_1,h_2\ge0: 1N∑n≤Nf1(n+h1)f2(n+h2)→0\frac1N\sum_{n\le N}f_1(n+h_1)f_2(n+h_2)\to0. Also for fixed nonproportional affine forms and in fixed residue classes. Functions may be complex and need not be completely multiplicative. Qualitative only, no rate; the kk-point Elliott conjecture for k≥3k\ge3 is not addressed.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single manuscript, dated September 24, 2026. The release also publishes an abridged summary of the model's reasoning for this family (reasoning_traces/ordinary-two-point-correlations.pdf).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.2 and Corollary 1.3 were read against the binary corrected Elliott conjecture with ordinary averages. The challenges are not in the formalization catalogue (lean/formalization.yaml); lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.json and OrdinaryElliott.json exist with solution modules OAI.NumberTheory.TwoPointCorrelations.FinalMain and OAI.NumberTheory.OrdinaryCorrelations.Elliott.Main present at the pinned commit, permitted axioms propext, Quot.sound and Classical.choice. The statements OAI.OrdinaryCorrelations.binary_corrected_elliott and OAI.OrdinaryTwoPointCorrelations.binary_corrected_elliott and affine_corrected_elliott were read: multiplicative one-bounded f1, f2, one of them with inf over |t| <= N of the distance to chi(n) n^{it} tending to infinity for every Dirichlet character, imply the shifted (and affine) correlation averages tend to 0. That is the headline. Not rebuilt here. The paper makes no rate claim for general functions.

Sources

Changelog1 change

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