Short Sums of the Liouville Function over Function Fields

TL;DR

This paper investigates the behavior of the Liouville function over function fields, establishing bounds for short sums and extending techniques from integer settings to function fields.

Contribution

It proves a new bound for short sums of the Liouville function over function fields, adapting methods from integer cases to the function field context.

Findings

Bound for the average squared sum of the Liouville function over short intervals.

Extension of integer-based methods to the function field setting.

Provides asymptotic bounds involving parameters q, N, and h.

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}.