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

Abstract

We examine correlations of the Möbius 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} μ(f)χ(Q(f)) \end{equation} for an additive character $\chi$ over ${\mathbb{F}}_q$ and a polynomial $Q\in {\mathbb{F}}_q[x_0,\dots ,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\dots ,x_{n-1}$ 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 $O(q^{-c^{\prime }n}$) for some $c^{\prime }>0$ 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.

24 pages, 1 figure. Second version optimises the saving in the linear case. Third version uses the proof by Hosseini and Lovett of the author's conjecture to make the main result unconditional

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