Hardy-Littlewood tuple conjecture over large finite field

TL;DR

This paper proves a function field analog of the Hardy-Littlewood conjecture, showing that the number of polynomials with certain irreducibility properties over large finite fields asymptotically matches a predicted formula.

Contribution

It establishes the first asymptotic result for the Hardy-Littlewood conjecture over large finite fields, extending the twin prime conjecture to a broad class of polynomial tuples.

Findings

Asymptotic formula (q,n;a) q q^n/n^r as q tends to infinity

Valid for fixed n,r and variable finite fields

Generalizes twin prime conjecture to polynomial setting over finite fields

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).