On the distribution of divisors of monic polynomials over function fields

TL;DR

This paper extends classical number theory results to function fields, analyzing the distribution of divisors and irreducible factors of monic polynomials over finite fields using an adapted Selberg-Delange method.

Contribution

It introduces a novel extension of the Selberg-Delange method to count monic polynomials with specified irreducible factors and divisor distributions in residue classes.

Findings

Count of monic polynomials with t irreducible factors

Distribution of divisors in residue classes

Extension of classical theorems to function fields

Abstract

This paper deals with function field analogues of famous theorems of Laudau which counted the number of integers which have $t$ prime factors and R. Hall which researched the distribution of divisors of integers in residue classes.\;We extend the Selberg-Delange method to handle the following problems.\;The number of monic polynomials with degree $n$ have $t$ irreducible factors;\;The number of monic polynomials with degree $n$ in some residue classes have $t$ irreducible factors and the residue classes distribution of divisors of monic polynomial.\;