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:

$ \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 $ \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
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