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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

peter
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    In general, we say "function field" analogue. I don't know a good survey, but maybe concerning objects this could be a good start. See also this question: https://mathoverflow.net/questions/1367/global-fields-what-exactly-is-the-analogy-between-number-fields-and-function-fi?noredirect=1&lq=1, especially this reference. – Watson Apr 14 '20 at 14:10
  • See also the non-analogies: https://mathoverflow.net/questions/177234 – Watson Apr 14 '20 at 14:16
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    Arguably counting prime polynomials of degree $n$ is not really the right analogue of the prime number theorem. A better one might be prime polynomials with the leading $k$ coefficients fixed. This requires nonvanishing on $s=1$ for a Dirichlet $L$-function, and can be proven by the same kind of complex-analytic argument. (Or by one of the four different proofs of the Riemann hypothesis, of course - but none of them are as trivial as the "some combinatorial reasonings" you mention, which give the wrong impression IMO.) – Will Sawin Apr 14 '20 at 14:31

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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

LeoOwler
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