Prime polynomial values of quadratic functions in short intervals

TL;DR

This paper proves a function field version of the Bateman-Horn conjecture for prime polynomial values of quadratic functions in short intervals, demonstrating key analytic number theory techniques in finite fields.

Contribution

It establishes the function field analogue of the Bateman-Horn conjecture in short intervals, including new results on Mobius sum cancellations and Chowla sum correlations.

Findings

Confirmed square root cancellation in Mobius sums implies similar cancellation in Chowla sums

Derived function field analogs of prime polynomial counts in short intervals

Extended analytic techniques to finite fields for prime polynomial distribution

Abstract

In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short intervals. To this end we further work out on function field analogs of cancellation of Mobius sums and its correlations(Chowla type sums) and confirm that square root cancellation in Mobius sums is equivalent to square root cancellation in Chowla type sums.