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Math-2789 · mathematics · Number theory

Ordinary binary affine Liouville correlations

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $\Omega(n)$ count prime factors with multiplicity and put $\lambda(n)=(-1)^{\Omega(n)}$ for positive integers $n$. Fix integers $a_1,a_2\ge1$ and $b_1,b_2\ge0$ with $a_1b_2-a_2b_1\ne0$. Must \[ \sum_{1\le n\le X}\lambda(a_1n+b_1)\lambda(a_2n+b_2)=o(X) \qquad(X\to\infty) \] hold through all real cutoffs $X$?

Short English statement

Must the ordinary average of the Liouville product along every such pair of distinct integer affine forms tend to zero?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2789-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2789

The source treats the same fixed positive slopes, nonnegative offsets and nonproportionality condition as this question, with no coprimality restriction. Its estimate holds through all real $X\ge3$ and saves a positive power of $\log X$. Dividing by $X$ therefore gives the requested ordinary-average limit zero, with additional quantitative decay. Higher-order Liouville correlations are a separate question.

Ordinary binary affine Liouville correlations

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

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