A cubic nonconventional ergodic average with M\"obius and Liouville weight

TL;DR

This paper proves that certain weighted ergodic averages involving M"obius and Liouville functions converge to zero almost surely, leading to new results on the decay of their self-correlations.

Contribution

It establishes the almost sure convergence of cubic nonconventional ergodic averages with M"obius and Liouville weights, a novel result in ergodic theory and number theory.

Findings

Cubic nonconventional ergodic averages with M"obius and Liouville weights converge to zero almost surely.

Cesàro means of self-correlations of M"obius and Liouville functions tend to zero.

Certain moving averages of these self-correlations also converge to zero.

Abstract

It is shown that the cubic nonconventional ergodic average of order 2 with M\"obius and Liouville weight converge almost surely to zero. As a consequence, we obtain that the Ces\`aro mean of the self-correlations and some moving average of the self-correlations of M\"obius and Liouville functions converge to zero.