Ergodicity of the Liouville system implies the Chowla conjecture

TL;DR

This paper demonstrates that ergodic properties of the Liouville function imply the truth of the Chowla conjecture, connecting ergodic theory with deep conjectures in number theory.

Contribution

It shows that ergodicity of the Liouville function suffices to prove the Chowla conjecture, weakening previous assumptions and linking ergodic theory to number theory conjectures.

Findings

Liouville function's ergodicity implies Chowla conjecture

Results relate to Elliott's conjecture on multiplicative functions

Uses ergodic theory, uniformity seminorms, and nilmanifold equidistribution

Abstract

The Chowla conjecture asserts that the values of the Liouville function form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory it asserts that the Liouville function is generic for the Bernoulli measure on the space of sequences with values plus or minus one. We show that these statements are implied by the much weaker hypothesis that the Liouville function is generic for an ergodic measure. We also give variants of this result related to a conjecture of Elliott on correlations of multiplicative functions with values on the unit circle. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary decomposition results involving uniformity seminorms, and qualitative equidistribution results on nilmanifolds.