Prime polynomial values of linear functions in short intervals

TL;DR

This paper proves a function field analogue of a major number theory conjecture, establishing an asymptotic formula for prime values of multiple linear functions in short intervals over large finite fields.

Contribution

It introduces a new function field framework that confirms a combined conjecture related to primes in various classical settings, extending previous number theory results.

Findings

Established an asymptotic formula for prime values of linear functions

Confirmed a conjecture in the function field setting analogous to classical number theory

Extended understanding of prime distributions in finite fields

Abstract

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.