Hardy-Littlewood tuple conjecture over large finite field

Abstract

We prove the following function field analog of the Hardy-Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n,r be positive integers and q an odd prime power. For distinct polynomials a_1, ..., a_r over F_q of degree <n let \pi(q,n;a) be the number of monic polynomials f over F_q of degree n such that f+a_1, ..., f+a_r are simultaneously irreducible. We prove that \pi(q,n;a) asymptotically equals q^n/n^r as q tends to infinity on odd prime powers and n,r are fixed (the tuple a1,...,a_r need not be fixed).