Square-root cancellation for sums of factorization functions over squarefree progressions in $\mathbb F_q[t]$

We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.

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