Correlation of arithmetic functions over $\mathbb{F}_q[T]$

Abstract

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $\Delta$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $\Delta$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $\Delta$. This allows us to obtain additional saving from equidistribution results for $L$-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the M\"{o}bius function and divisor functions.