Short Sums of the Liouville Function over Function Fields

TL;DR

This work analyzes the short-interval behavior of the Liouville function in the function-field setting. By transferring ideas from the integer case to $\\mathbb{F}_q[t]$ and employing a function-field $L^2$ mean value theorem, an involution on polynomials, and a Ramaré-type identity, the authors obtain a variance bound for sums of $\\lambda$ over short intervals: for fixed $q$ and $h \ll \sqrt{N}$ with $h(N) \to \infty$, $\\frac{1}{q^N}\\sum_{G_0\\in \\mathcal{M}_N} |\\sum_{G\\in \\mathcal{I}_h(G_0)} \\lambda(G)|^2 \\ll_q \\\frac{N^5}{h^2} q^{h}$. This parallels square-root cancellation results in the integer setting (via Matomäki–Radziwiłł) and demonstrates strong average cancellations in a fixed-$q$ regime, highlighting how function-field tools can sharpen variance bounds and connect to large-$q$ analogues. The approach integrates an involutive transformation, orthogonality of even characters, and a decomposition into smooth versus non-smooth polynomials to control contributions. Overall, the result advances understanding of Liouville-type multiplicative functions on function fields and their short-interval fluctuations.

Abstract

Let $λ$ 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)}λ(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 Matomäki and Radziwiłł.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Remark
• Lemma 1.3
• proof
• Theorem 1.4
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4