It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem
$$\#\{\text{prime numbers}\leq x\} \sim \frac{x}{\log x}, \;\;\,\; x \to +\infty,$$
requires complex analysis and the study of the Riemann zeta function (or the elementary, albeit complicated, machinery of Selberg), while, on the other hand, the proof of
$$\#\{\text{irreducible monic polynomials in $\mathbb{F}\_q[x]$ of degree} \leq n\} \sim \frac{q^n}{\log\_q q^n}$$
requires only some combinatorial reasonings (which also gives an error term of the size of the one expected for the error term of the prime numbers counting function under the Riemann's hypothesis).


Do you know where I can find a survey or some other kind of collection of these finite field analog? Thanks



Reference / Survey for finite field analog number theory
One of the best references I am aware of, is the book by Rosen ‘Number theory in function fields’ <https://www.springer.com/gp/book/9780387953359>.


A good survey, which also concerns the important connection with random matrix theory, is the following paper by Rudnick, although you won’t find proofs there: <http://www.math.tau.ac.il/~rudnick/papers/ICM-Proceedings-Rudnickv5.pdf>


