Odd order cases of Chowla's averages over arbitrary rectangles

Let $\lambda$ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k$. that is, for every fixed even integer $k\ge 2$, we prove that $$\underset{\min \{M,N\}\to \mathrm{\infty }}{\lim }\frac{1}{MN}\sum _{m=1}^M\sum _{n=1}^N\lambda (m)\lambda (m+a_{1}n)\cdots \lambda (m+a_{k}n)=0.$$ We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.