Cancellation in correlations of the Möbius function over function fields

Question

Let $p$ be an odd prime and $q$ a power of $p$. For a polynomial $f \in \mathbb{F}_q[T]$, let $\mu(f)$ be the Möbius function of $f$. For a positive integer $d$, let $M_d$ be the set of monic polynomials in $\mathbb{F}_q[T]$ of degree $d$.

For fixed distinct, nonzero polynomials $h_1, \dots, h_k \in \mathbb{F}_q[T]$, what is the best known upper bound for the sum $$\sum_{f \in M_d} \mu(f + h_1) \dotsm \mu(f + h_k)?$$ In particular, for which values of $q$ (as a function of $p$ and $k$) can one obtain cancellation, i.e., an upper bound of $o(|M_d|)$ as $d \to \infty$?

Answer 1

One can obtain cancellation for $q > p^2k^2 e^2$ by a result of myself and Mark Shusterman.

Indeed we obtain a bound of $O( q^{d (1 - \frac{\beta}{p} ) })$ for any $\beta < \frac{1}{2} - \frac{\log (pne)}{\log q} $.

A reference is On the Chowla and twin primes conjectures over $\mathbb F_q[T]$, Annals of Mathematics 196 (2022), Will Sawin and Mark Shusterman.