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 Cesàro 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 $N \geq 2$, $$\frac1{N}\sum_{m=1}^{N}\Big|\frac1{N}\sum_{n=1}^{N} \bnu(n) \bnu(n+m)\Big| \leq \frac{C}{\log(N)^ε},$$ 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.

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