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The genuine prime number theorem says that there are about prime numbers between 1 and N. The prime number theorem in function fields is rather about the number
of irreducible monic polynomials of degree
in the polynomial ring
on a finite field
. It is remarkable that the result is of the same shape
, which is
if you let
be the number of monic polynomials of degree
. But the proof, as often with the polynomial analogues of number theory, is much neater and even yields a nice exact formula for
First proof
This one requires a good deal of knowledge of the theory finite fields.
We prove that is the product
of all monic irreducible polynomials whose degree divide
, which immediately provides the above formula by comparison of the degrees. To prove this, notice that the roots of
are all the elements of the extension
of
, so that
, as an element of
can be factored as
. On the other hand, if
is an irreducible polynomial of degree
, it splits into linear factors in
. So
has a root
and all its other roots are images of
under the Frobenius element, more precisely,
so that (at first sight in
, but also in
because they are both in
– it is well known that the gcd is preserved under field extensions). Moreover, all monic irreducible
are coprimes, so that their product still divides
. Finally, we need to prove the converse divisibility, either in
or in
, which is equivalent. So let’s do it in the latter; take a factor
with
. Let
be the order of the Frobenius element on
, so that
and no smaller integer has this property; then
which is indeed an irreducible of degree
, hence a factor of
. Given the factorization of
and the fact that the factors
are coprime, their product
still divides
Second proof
This one proves exactly what we want and not more and uses only power series. We need the notation .
Let
be the zeta function (for s of real part >1). It is easy to see that by looking at the sum expression, and
by looking at the product expression. Putting
, both sides become nice power series so that we get the equality
and doing classical operations (logarithmic derivation and expansion in power series followed by termwise identification in particular) on both sides yield the result.
This proof seems to use about zero input of algebra, so its success looks like a miracle. The only step where a (tiny) bit of knowledge seems to be used is the transformation of the sum into an Euler product – it needs the existence of unicity of the decomposition of a monic polynomial into product of irreducibles.
A kind of sieve method ?
It could be reasonable to hope that the sieve of Eratosthenes would work better in the realm of polynomial rings. We’ll try to adapt it here. Notice that in this case we have count primes up to a certain large degree, and not of exactly certain large degree (both quantities are far from equivalent in fact !).
Let’s define to be the number of monic irreducible (let’s say primes) polynomials of degree at most n, and
. We also use Moebius’ function for monic polynomials, also defined in terms of the number of prime factors. The Moebius inversion formula
still holds in the polynomial world. We also write
Thus
and if we forget about we obtain exactly
this looks quite nice, but the error term, just like in the integer case, can’t easily be bounded by anything better than which we know will be vastly bigger than the main term. By the way this product can very easily be shown to be
but the exact asymptotic is not any easier than the PNT itself… So this sieve seems to have very little interest, given the convenience of the other methods.
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