12 1. Elementary Prime Number Theory, I
It follows that
π(N)
log N
log(1 + log N/ log 2)
.
Taking some care to estimate the denominator, we obtain the lower bound
π(N) (1 + o(1))
log N
log log N
,
which tends to infinity. Similar proofs of Theorem 1.1 have been given
by Thue (1897), Auric (1915), Schnirelmann [Sch40, pp. 44–45], Chernoff
[Che65], and Rubinstein [Rub93]. See also Exercise 17.
6. Sledgehammers!
In the spirit of the saying, “nothing is too simple to be made complicated,”
we finish off the first half of this chapter with two proofs of Theorem 1.1
that dip into the tool chest of higher mathematics.
The following “topological proof” is due to Furstenberg ([Fur55]):
Proof. We put a topology on Z by taking as a basis for the open sets
all arithmetic progressions, infinite in both directions. (This is permissible
since the intersection of two such progressions is either empty or is itself an
arithmetic progression.) Then each arithmetic progression is both open and
closed: it is open by choice of the basis, and it is closed since its complement
is the union of the other arithmetic progressions with the same common
difference. For each prime p, let Ap = pZ, and define A :=
p
Ap. The
set {−1, 1} = Z \ A is not open. (Indeed, each open set is either empty or
contains an arithmetic progression, so must be infinite.) It follows that A is
not closed. On the other hand, if there are only finitely many primes, then
A is a finite union of closed sets, and so it is closed.
Our next proof, due to L. Washington (and taken from [Rib96]) uses
the machinery of commutative algebra. Recall that a Dedekind domain is
an integral domain R with the following three properties:
(i) R is Noetherian: if I1 I2 I3 · · · is an ascending chain of
ideals of R, then there is an n for which
In = In+1 = In+2 = · · · .
(ii) R is integrally closed: if K denotes the fraction field of R and
α K is the root of a monic polynomial with coefficients in R,
then in fact α R.
(iii) Every nonzero prime ideal of R is a maximal ideal.
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