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Math-2775 · mathematics · Number theory

Corrected binary Elliott for ordinary averages

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let f1,f2 be multiplicative functions of modulus at most one. Suppose at least one f satisfies, for every fixed Dirichlet character chi, $inf_{|t|<=N}$ $sum_{p<=N}$ (1-Re(f(p) $conjugate(chi(p)p^{it})))/p$ $->infinity$. For distinct fixed nonnegative h1,h2, must (1/N) $sum_{n<=N}$ f1(n+h1)f2(n+h2) tend to zero?

Short English statement

Let f1,f2 be multiplicative functions of modulus at most one. Suppose at least one f satisfies, for every fixed Dirichlet character chi, $inf_{|t|<=N}$ $sum_{p<=N}$ (1-Re(f(p) $conjugate(chi(p)p^{it})))/p$ $->infinity$. For distinct fixed nonnegative h1,h2, must (1/N) $sum_{n<=N}$ f1(n+h1)f2(n+h2) tend to zero?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2775-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2775

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 007; 2026-09-24).

Corrected binary Elliott for ordinary averages

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.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

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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