Short Sums of the Liouville Function over Function Fields

Abstract

Let $\lambda$ denote the Liouville function for function fields. We prove that for a fixed $q$, given $h \ll \sqrt{N}$ and $h(N) \to \infty$ arbitrarily slowly as $N \to \infty$, then \begin{equation*} \frac{1}{q^N}\sum_{G_0 \in \mathcal{M}_N}|\sum_{G \in \mathcal{I}_{h}(G_0)}\lambda(G)|^2 \ll_q \frac{N^5}{h^2}q^{h}. \end{equation*} The proof follows a similar method of an analogous case in the integer setting developed by Chinis, adapting methods originally developed by Matom\"aki and Radziwi{\l}{\l}.