Correlation of arithmetic functions over FqT.

For a fixed polynomial Δ , we study the number of polynomials f of degree n over F q such that f and f + Δ are both irreducible, an 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 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. [ABSTRACT FROM AUTHOR]