Linear and quadratic uniformity of the M\"obius function over $\mathbb{F}_q[t]$

TL;DR

This paper investigates the correlation of the Möbius function over polynomial rings over finite fields with linear and quadratic phases, establishing bounds that suggest significant cancellation and randomness similar to the integer case.

Contribution

The paper provides new bounds on Möbius function correlations with linear and quadratic phases over $\,\mathbb{F}_q[t]$, extending understanding of its pseudorandomness in function fields.

Findings

Bound of $O_\\epsilon(q^{(-1/4+\\epsilon)n})$ for linear phases

Bound of $O(q^{-n^c})$ for quadratic phases

Improved bounds for Hankel quadratic forms

Abstract

We examine correlations of the M\"obius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} \mu(f)\chi(Q(f)) \end{equation} for an additive character $\chi$ over $\mathbb{F}_q$ and a polynomial $Q\in\mathbb{F}_q[x_0,\ldots,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\ldots, x_{n-1}$ of $f=\sum_{i< n}x_i t^i$. Like in the integers, it is reasonable to expect that, due to the random-like behaviour of $\mu$, such sums should exhibit considerable cancellation. In this paper we show that the above correlation is bounded by $O_\epsilon \left( q^{(-\frac{1}{4}+\epsilon)n} \right)$ for any $\epsilon >0$ if $Q$ is linear and $O \left( q^{-n^c} \right)$ for some absolute constant $c>0$ if $Q$ is quadratic. The latter bound may be reduced to $O(q^{-c'n}$) for some $c'>0$ when $Q(f)$ is a linear form in the coefficients of $f^2$, that is, a Hankel quadratic form, whereas for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.