Odd order cases of Chowla's averages over arbitrary rectangles

Abstract: Let $λ$ 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 $k\geq 2$, we prove that [ \lim_{\min{M,N}\to \infty}\frac{1}{MN}\sum_{m=1}^M\sum_{n=1}^Nλ(m)λ(m+a_1n)\cdotsλ(m+a_kn)=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.