Correlation of arithmetic functions over 𝔽𝑞[𝑇]

For a fixed polynomial 𝛥, we study the number of polynomials f of degree n over 𝔽𝑞 such that f and 𝑓+𝛥 are both irreducible, an 𝔽𝑞[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 𝛥 in a manner which is consistent with the Hardy–Littlewood Conjecture. We obtain a saving of q if we consider monic polynomials only and 𝛥 is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in 𝛥. 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 Möbius function and divisor functions.