Counting irreducible polynomials over finite fields

Abstract

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem:

$$ \pi (x) = \frac{q} {{q - 1}}\frac{x} {{\log _q x}} + \frac{q} {{(q - 1)^2 }}\frac{x} {{\log _q^2 x}} + O\left( {\frac{x} {{\log _q^3 x}}} \right),x = q^n \to \infty $$

where π(x) denotes the number of monic irreducible polynomials in F _q [t] with norm ⩽ x.