Odd order cases of the logarithmically averaged Chowla conjecture

TL;DR

This paper provides a new, shorter proof for the odd order cases of the logarithmically averaged Chowla conjecture, eliminating the need for ergodic theory used in previous proofs.

Contribution

It introduces a novel proof technique for the odd order cases of the conjecture, simplifying the argument and removing reliance on ergodic theory.

Findings

Established the odd order cases of the conjecture

Provided a proof that avoids ergodic theory

Simplified the understanding of correlations in the Liouville function

Abstract

A famous conjecture of Chowla states that the Liouville function $\lambda(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.