Hardy–Littlewood Tuple Conjecture Over Large Finite Fields

We prove the following function field analog of the Hardy–Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n and r be positive integers and q an odd prime power. For a tuple of distinct polynomials
${\rm a}=(a_1,...,a_r)\in \mathbb F_q[t]^r$
of degree $f\in\mathbb F_q[t]$
of degree n such that f+a
1
,…,f+a
r
are simultaneously irreducible. We prove that
$\pi(q,n;{\rm a})\sim\frac{q^n}{n^r}$
as
$q\rightarrow \infty$
and n,r fixed.