# Article

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Keywords:
finite fields; distribution of irreducible polynomials; residue
Summary:
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 $\le x$.
References:
[1] Kruse, M., Stichtenoth, H.: Ein Analogon zum Primzahlsatz fur algebraische Functionenkoper. Manuscripta Math. 69 (1990), 219-221 German. DOI 10.1007/BF02567920 | MR 1078353
[2] Davenport, H.: Multiplicative Number Theory. Springer-Verlag New York (1980). MR 0606931 | Zbl 0453.10002

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