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

Abstract

Let $$q\geqslant 2$$ 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.