A cubic nonconventional ergodic average with multiplicative or Mangoldt   weights
  We show that the cubic nonconventional ergodic averages of any order with a bounded multiplicative function weight converge almost surely to zero provided that the multiplicative function satisfies a strong Daboussi-Delange condition. We further obtain that the Ces\`{a}ro mean of the self-correlations and some moving average of the self-correlations of such multiplicative functions converge to zero. Our proof gives, for any $N \geq 2$, $$\frac1{N}\sum_{m=1}^{N}\Big|\frac1{N}\sum_{n=1}^{N} \bnu(n) \bnu(n+m)\Big| \leq \frac{C}{\log(N)^{\epsilon}},$$ and $$\frac1{N^2}\sum_{n,p=1}^{N}\Big|\frac1{N}\sum_{m=1}^{N} \bnu(m) \bnu(n+m)\bnu(m+p)\bnu(n+m+p)\Big| \leq \frac{C}{\log(N)^{\varepsilon}},$$ where $C,\varepsilon$ are some positive constants and $\bnu$ is a bounded multiplicative function satisfying a Daboussi-Delange condition with logarithmic speed. We further establish that the cubic nonconventional ergodic averages of any order with Mangoldt weight converge almost surely provided that all the systems are nilsystems. 
el Houcein el Abdalaoui and XiangDong Ye
math.DS math.CO math.CV math.NT math.PR