A cubic nonconventional ergodic average with Möbius and Liouville weight

Abstract

It is shown that the cubic nonconventional ergodic average of order 2 with Möbius and Liouville weight converge almost surely to zero. As a consequence, we obtain that the Cesà ro mean of the self-correlations and some moving average of the self-correlations of Möbius and Liouville functions converge to zero.

In this version, we put in the surface our main result on the Cesaro mean of the auto-correlation of Möbius and Liouville which is related to the very recent results of K. Matomäki and M. Radziwiłł, and K. Matomäki, M. Radziwiłł and T. Tao