Geodesic
Counting irreducible polynomials over finite fields
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 881-886
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In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: \[ \pi (x)= \frac q{q - 1}\frac x{{\log _q x}}+ \frac q{(q - 1)^2}\frac x{{\log _q^2 x}}+O\Bigl (\frac {x}{{\log _q^3 x}}\Bigr ),\quad x=q^n\rightarrow \infty \] where $\pi (x)$ denotes the number of monic irreducible polynomials in $F_q [t]$ with norm $ \le x$.
In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: \[ \pi (x)= \frac q{q - 1}\frac x{{\log _q x}}+ \frac q{(q - 1)^2}\frac x{{\log _q^2 x}}+O\Bigl (\frac {x}{{\log _q^3 x}}\Bigr ),\quad x=q^n\rightarrow \infty \] where $\pi (x)$ denotes the number of monic irreducible polynomials in $F_q [t]$ with norm $ \le x$.
Classification : 11T55
Keywords: finite fields; distribution of irreducible polynomials; residue 
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     author = {Wang, Qichun and Kan, Haibin},
     title = {Counting irreducible polynomials over finite fields},
     journal = {Czechoslovak Mathematical Journal},
     pages = {881--886},
     year = {2010},
     volume = {60},
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     zbl = {1224.11086},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a18/}
}
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Wang, Qichun; Kan, Haibin. Counting irreducible polynomials over finite fields. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 881-886. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a18/

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[2] Davenport, H.: Multiplicative Number Theory. Springer-Verlag New York (1980). | MR | Zbl