Möbius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields

Abstract

We establish cancellation in short sums of certain special trace functions over ${\mathbb{F}}_q[u]$ below the Pólya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the ${\mathbb{F}}_q[u]$-analog of Chowla's conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in ${\mathbb{F}}_q[u]$.