The function field Sath\'e-Selberg formula in arithmetic progressions and `short intervals'

TL;DR

This paper develops a function field analogue of the Sathé-Selberg formula, providing asymptotic counts of polynomials with specific properties in arithmetic progressions and short intervals, leveraging Weil's Riemann hypothesis for improved results.

Contribution

It introduces a new method for counting polynomials with fixed irreducible factors in function fields, extending Selberg's approach to arithmetic progressions and short intervals with enhanced bounds.

Findings

Derived asymptotic formulas for square-free polynomials with k factors

Extended counting methods to arithmetic progressions and short intervals

Achieved better bounds using Weil's Riemann hypothesis

Abstract

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which $q$ tends to infinity.