The two-point Chowla conjecture (ordinary averages)

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. Chowla's conjecture predicts that $\sum_{n\le X}\lambda(n+h_1)\cdots\lambda(n+h_k)=o(X)$ for every fixed set of distinct shifts. Its two-point case asks whether the parities of $\Omega(n)$ and $\Omega(n+h)$ become uncorrelated. Matomaki and Radziwill obtained a bound $(1-\delta(h))X$ , Matomaki-Radziwill-Tao proved cancellation on average over shifts, and Tao proved the logarithmically averaged version, with quantitative log-averaged savings by Helfgott-Radziwill and Pilatte. Does $\sum_{n\le X}\lambda(n)\lambda(n+h)=o(X)$ hold for every fixed $h\ge1$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Analytic number theory; correlations of multiplicative functions
Posed by: Sarvadaman Chowla, The Riemann Hypothesis and Hilbert's Tenth Problem (Gordon and Breach, 1965)
Year posed: 1965
Years open: 61y
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: 58 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

There is an absolute $c>0$ such that for all fixed integers $a_1,a_2\ge1$ , $b_1,b_2\ge0$ with $a_1b_2\ne a_2b_1$ , $\left|\sum_{n\le X}\lambda(a_1n+b_1)\lambda(a_2n+b_2)\right|\le C_{a,b}X/(\log X)^c$ for every real $X\ge3$ . With $a_1=a_2=1$ , $b_1=0$ , $b_2=h$ this is the ordinary two-point Chowla conjecture with a power-of-log saving. Constants are ineffective, and coefficients may not grow with $X$ . Nothing is proved for three or more points.

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 1.1 was read against the two-point case of Chowla's 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 statement OAI.OrdinaryTwoPointCorrelations.liouville_log_saving was read: an absolute c > 0 such that for natural a1, a2 >= 1, b1, b2 with a1 b2 != a2 b1 there is C > 0 with |sum_{n <= X} lambda(a1 n + b1) lambda(a2 n + b2)| <= C X / (log X)^c for all real X >= 3. That covers the headline, including every shift h >= 1. Not rebuilt here. The paper says the constants need not be effective and no uniformity in the forms is claimed.

The two-point Chowla conjecture (ordinary averages)

What was actually shown

What the AI did

Verification

Sources

Discussion