A dynamical generalization of Chowla's conjecture on average

Let $k\ge 1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda (n+h_1),\dots ,\lambda (n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1,\dots ,h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla’s conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter’s theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla’s conjecture along the primes on average as well.