Prime Numbers

16 Chapter 1 PRIMES!
Let’s hear it for heuristic reasoning! Very recently P. Sebah found
16
π2 10 = 10304195697298,
as enunciated in [Gourdon and Sebah 2004].
As strong as the numerical evidence may be, we still do not even know
whether there are infinitely many pairs of twin primes; that is, whether π2(x)is
unbounded. This remains one of the great unsolved problems in mathematics.
The closest we have come to proving this is the theorem of Chen Jing-run
in 1966, see [Halberstam and Richert 1974], that there are infinitely many
primes p such that either p + 2 is prime or the product of two primes.
A striking upper bound result on twin primes was achieved in 1915 by
V. Brun, who proved that
2
ln ln x
π2(x) =O x
, (1.8)
ln x
and a year later he was able to replace the expression ln ln x with 1
(see [Halberstam and Richert 1974]). Thus, in some sense, the twin prime
conjecture (1.7) is partially established. From (1.8) one can deduce (see
Exercise 1.50) the following:
Theorem 1.2.1 (Brun). The sum of the reciprocals of all primes belonging
to some pair of twin primes is finite, that is, if P2 denotes the set of all primes
p such that p +2 is also prime, then
1 1
+ < ∞.
p p +2
p∈P2
(Note that the prime 5 is unique in that it appears in two pairs of twins,
and in its honor, it gets counted twice in the displayed sum; of course, this
has nothing whatsoever to do with convergence or divergence.) The Brun
theorem is remarkable, since we know that the sum of the reciprocals of all
primes diverges, albeit slowly (see Section 1.1.5). The sum in the theorem,
namely
B ′ =(1/3+1/5) + (1/5+1/7) + (1/11 + 1/13) + ···,
is known as the Brun constant. Thus, though the set of twin primes may well
be infinite, we do know that they must be significantly less dense than the
primes themselves.
An interesting sidelight on the issue of twin primes is the numerical
calculation of the Brun constant B ′ . There is a long history on the subject,
with the current computational champion being Nicely. According to the
paper [Nicely 2004], the Brun constant is likely to be about
B ′ ≈ 1.902160583,