An averaged Chowla and Elliott conjecture along independent polynomials

TL;DR

This paper proves an averaged version of the Chowla and Elliott conjectures for multiplicative functions with shifts given by independent polynomials, using ergodic theory and recent results on mean values of multiplicative functions.

Contribution

It introduces a novel ergodic approach to average correlations of multiplicative functions along polynomial shifts, extending previous results to a broader polynomial setting.

Findings

Established an averaged correlation result for multiplicative functions along independent polynomial shifts.

Connected ergodic theory with number theory to analyze polynomial shift correlations.

Derived new patterns in the range of arithmetic sequences along polynomial shifts.

Abstract

We generalize a result of Matom\"aki, Radziwi{\l}{\l}, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative functions, with shifts given by independent polynomials in several variables. A new feature is that we recast the problem in ergodic terms and use a multiple ergodic theorem to prove it; its hypothesis is verified using recent results by Matom\"aki and Radziwi{\l}{\l} on mean values of multiplicative functions on typical short intervals. We deduce several consequences about patterns that can be found on the range of various arithmetic sequences along shifts of independent polynomials.