Result 007, Number theory

Ordinary two-point correlations and the corrected Elliott conjecture

Proves the ordinary two-point Chowla conjecture, with a bound O(X/(log⁡X)c)O(X/(\log X)^c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times nitn^{it} for ∣t∣≤X|t|\le X.

Lean formalization Reasoning summary (PDF) Proof

The bigger picture

Why it matters

The Liouville function records whether an integer has an even or odd number of prime factors, counted with repetition. The manuscript reports cancellation in products of two such signs, sharpening the picture of randomness in arithmetic.

What changes?

For two fixed affine forms, an + b, that are not proportional, the manuscript bounds the Liouville correlation sum up to X by a constant times X divided by (log X) to an absolute positive power, at every cutoff. It also claims vanishing ordinary two-point averages for complex multiplicative functions of magnitude at most one, assuming one original factor uniformly avoids imitating each fixed Dirichlet character times n to the power it, for t of magnitude at most X.

What does that help mathematicians do?

Here a Dirichlet character is a periodic multiplicative pattern, and n to the power it supplies a smoothly varying complex phase. The uniform avoidance condition is called nonpretentiousness. Under it, the qualitative conclusion also holds within fixed residue classes and along fixed nonproportional affine forms. Researchers could therefore deduce cancellation even after restricting inputs to a fixed remainder. Crucially, the hypothesis excludes imitation of both periodic patterns and their phase-modified versions, not merely periodicity alone.

Are there practical applications?

Its immediate value is foundational: multiplicative functions respect multiplication of coprime integers, yet these results concern their behavior along additive patterns. Ordinary averages give each input equal weight. The reported bounds would provide quantitative control of Liouville's two-point correlations, not a guarantee of independence for larger collections of values.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Ordinary two-point correlations of multiplicative functions

September 24, 2026 87 pages

We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026,
  author = {{OpenAI}},
  title = {{Ordinary two-point correlations of multiplicative functions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf}{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/007.md.

Ordinary two-point correlations and the corrected Elliott conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves ordinary two-point cancellation for multiplicative functions. For the Liouville function on every fixed pair of nonproportional affine forms, the correlation sum up to XX is O(X/(log⁡X)c)O(X/(\log X)^c) for an absolute c>0c>0, with the implied constant depending on the forms.

For two one-bounded multiplicative functions, if at least one is uniformly nonpretentious against all Dirichlet-character twists with frequency ∣t∣≤N|t|\le N, then their shifted and nonproportional affine correlation sums divided by NN tend to zero. These are ordinary averages, with no logarithmic averaging.

Comparator links

Result Comparator statement
Ordinary Elliott cancellation under uniform nonpretentiousness OrdinaryElliott.lean
Ordinary Liouville and multiplicative two-point correlations OrdinaryTwoPointCorrelations.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.