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Math-88-A2 · revision r1 · on Math-88

OpenAI claim summary for Math-88

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The cancellation theorem controls two Liouville factors on fixed affine forms. The problem asks for arbitrarily many factors, so higher correlations cannot be inferred from this binary result.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Chowla conjecture on correlations of the Liouville function
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Chowla conjecture on correlations of the Liouville function

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 007; 2026-09-24), the asserted result is that ordinary two-point Liouville averages have a logarithmic power saving on fixed nonproportional affine forms, and bounded multiplicative functions satisfy the corrected binary Elliott assertion under uniform nonpretentiousness.

Mathematical statement of the claimed result

Manuscript 1.
For Liouville $\lambda(n)=(-1)^{\Omega(n)}$, there is claimed an absolute $c>0$ such that every fixed $a_1,a_2\ge1$, $b_1,b_2\ge0$ with $a_1b_2-a_2b_1\ne0$ has a constant $C$ satisfying \[ |\sum_{n\le X}\lambda(a_1n+b_1)\lambda(a_2n+b_2)|\le C X/(\log X)^c\quad(X\ge3). \] The coefficients need not be coprime, but are fixed as $X$ grows. A second claim concerns multiplicative $f_1,f_2:\mathbb N\to\{z:|z|\le1\}$, at least one satisfying, for every fixed Dirichlet character $\chi$, \[ \inf_{|t|\le N}\left(\sum_{p\le N}\frac{1-\operatorname{Re}(f(p)\overline{\chi(p)p^{it}})}p\right)^{1/2}\longrightarrow\infty. \] For any distinct fixed nonnegative $h_1,h_2$ the claimed ordinary, rather than logarithmically weighted, conclusion is \[ N^{-1}\sum_{n=1}^Nf_1(n+h_1)f_2(n+h_2)\longrightarrow0. \]

Outline of the argument

For manuscript 1 (family 007; 2026-09-24), centering decomposes the correlation into graph and rough-number contributions. Short-interval cancellation controls a Fourier multiplier, while a conditioned finite-cylinder sieve extracts enough graph constraints to contradict a persistent bias.

Formal verification

For manuscript 1 (family 007; 2026-09-24), OpenAI lists Lean statements for this assertion: ordinary two-point Liouville averages have a logarithmic power saving on fixed nonproportional affine forms, and bounded multiplicative functions satisfy the corrected binary Elliott assertion under uniform nonpretentiousness.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are OrdinaryElliott.lean, OrdinaryTwoPointCorrelations.lean.

Their named declarations include OAI.OrdinaryCorrelations.binary_corrected_elliott, OAI.OrdinaryTwoPointCorrelations.affine_corrected_elliott, OAI.OrdinaryTwoPointCorrelations.binary_corrected_elliott, OAI.OrdinaryTwoPointCorrelations.liouville_log_saving.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources