Prime Numbers

1.1 Problems and progress 13
the reciprocals of the primes contained within this arithmetic progression is
infinite.
This marvelous (and nontrivial) theorem has been given modern refinement.
It is now known that if π(x; d, a) denotes the number of primes in the residue
class a mod d that do not exceed x, then for fixed coprime integers a, d with
d>0,
π(x; d, a) ∼ 1 1
π(x) ∼
ϕ(d) ϕ(d)
x
ln x
1
∼ li (x). (1.5)
ϕ(d)
Here ϕ is the Euler totient function, so that ϕ(d) is the number of integers
in [1,d] that are coprime to d. Consider that residue classes modulo d that
are not coprime to d can contain at most one prime each, so all but finitely
many primes are forced into the remaining ϕ(d) residue classes modulo d, and
so (1.5) says that each such residue class modulo d receives, asymptotically
speaking, its fair parcel of primes. Thus (1.5) is intuitively reasonable. We
shall later discuss some key refinements in the matter of the asymptotic error
term. The result (1.5) is known as the “prime number theorem for residue
classes.”
Incidentally, the question of a set of primes themselves forming an
arithmetic progression is also interesting. For example,
{1466999, 1467209, 1467419, 1467629, 1467839}
is an arithmetic progression of five primes, with common difference d = 210. A
longer progression with smaller primes is {7, 37, 67, 97, 127, 157}. Itisamusing
that if negatives of primes are allowed, this last example may be extended to
the left to include {−113, −83, −53, −23}. See Exercises 1.41, 1.42, 1.45, 1.87
for more on primes lying in arithmetic progression.
A very recent and quite sensational development is a proof that there
are in fact arbitrarily long arithmetic progressions each of whose terms is
prime. The proof does not follow the “conventional wisdom” on how to attack
such problems, but rather breaks new ground, bringing into play the tools of
harmonic analysis. It is an exciting new day when methods from another area
are added to our prime tool-kit! For details, see [Green and Tao 2004]. It has
long been conjectured by Erdős and Turán that if S is a subset of the natural
numbers with a divergent sum of reciprocals, then there are arbitrarily long
arithmetic progressions all of whose terms come from S. Since it is a theorem
of Euler that the reciprocal sum of the primes is divergent (see the discussion
surrounding (1.19) and Exercise 1.20), if the Erdős–Turán conjecture is true,
then the primes must contain arbitrarily long arithmetic progressions. The
thought was that maybe, just maybe, the only salient property of the primes
needed to gain this property is that their reciprocal sum is divergent. Alas,
Green and Tao use other properties of the primes in their proof, leaving the
Erdős–Turán conjecture still open.
Green and Tao use in their proof a result that at first glance appears
to be useless, namely Szemerédi’s theorem, which is a weaker version of the