Ordinary two-point correlations and the corrected Elliott conjecture

Proves the ordinary two-point Chowla conjecture, with a bound 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 n^{it} for |t|\le X. (Lean)

Manuscript

Ordinary two-point correlations of multiplicative functions

September 24, 2026 · source · cite as \cite{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026}

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.

Result 007 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.