Two averaged dynamical generalizations of Chowla's conjecture

Let $k\ge1$ be an integer and let $λ$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $λ(n+h_1),\dots, λ(n+h_k)$ 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