Counting irreducible polynomials over finite fields

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi (x) = \frac{q} {{q - 1}}\frac{x} {{\log _q x}} + \frac{q} {{(q - 1)^2 }}\frac{x} {{\log _q^2 x}} + O\left( {\frac{x} {{\log _q^3 x}}} \right),x = q^n \to \infty $$\end{document} where π(x) denotes the number of monic irreducible polynomials in Fq[t] with norm ⩽ x.