On the distribution of polynomials having a given number of irreducible factors over finite fields

Abstract

Let $q⩾2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field ${\mathbb{F}}_q$. We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to $x$ with exactly $k$ prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when $k$ is a little larger than $\log n$ than what one would speculate from looking at the integer case.

arXiv comment25 pages, 3 figures. Update: This paper reproduces an earlier result of Hwang using a different approach