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 1

$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f<n}\unicode[STIX]{x1D707}(f)\unicode[STIX]{x1D712}(Q(f))\end{eqnarray}$$

for an additive character $\unicode[STIX]{x1D712}$ 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 $f=\sum _{i<n}x_{i}t^{i}$. As in the integers, it is reasonable to expect that, due to the random-like behaviour of $\unicode[STIX]{x1D707}$, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by $O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$ for any $\unicode[STIX]{x1D716}>0$ if $Q$ is linear and $O(q^{-n^{c}})$ 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.