Prime Numbers

1.2 Celebrated conjectures and curiosities 19
where C2 is the twin-prime constant of (1.6). The Brun method can be used
to establish that R2(n) isbig-O of the right side of (1.9) (see [Halberstam and
Richert 1974).
Checking (1.9) numerically, we have R2(10 8 ) = 582800, while the right
side of (1.9) is approximately 518809. One gets better agreement using the
asymptotically equivalent expression R2(n) defined as
R2(n) =2C2
n−2
2
dt
(ln t)(ln(n − t))
p|n,p>2
p − 1
, (1.10)
p − 2
which at n =10 8 evaluates to about 583157.
As with twin primes, [Chen 1966] also established a profound theorem on
the Goldbach conjecture: Any sufficiently large even number is the sum of a
prime and a number that is either a prime or the product of two primes.
It has been known since the late 1930s, see [Ribenboim 1996], that “almost
all” even integers have a Goldbach representation p + q, the “almost all”
meaning that the set of even natural numbers that cannot be represented
as a sum of two primes has asymptotic density 0 (see Section 1.1.4 for the
definition of asymptotic density). In fact, it is now known that the number of
exceptional even numbers up to x that do not have a Goldbach representation
is O x 1−c for some c>0 (see Exercise 1.41).
The Goldbach conjecture has been checked numerically up through 10 14
in [Deshouillers et al. 1998], through 4 · 10 14 in [Richstein 2001], and through
10 17 in [Silva 2005]. And yes, every even number from 4 up through 10 17 is
indeed a sum of two primes.
As Euler noted, a corollary of the assertion that every even number after
2 is a sum of two primes is the additional assertion that every odd number
after 5 is a sum of three primes. This second assertion is known as the
“ternary Goldbach conjecture.” In spite of the difficulty of such problems of
additive number theory, Vinogradov did in 1937 resolve the ternary Goldbach
conjecture, in the asymptotic sense that all sufficiently large odd integers n
admit a representation in three primes: n = p + q + r. It was shown in 1989 by
Chen and Y. Wang, see [Ribenboim 1996], that “sufficiently large” here can
be taken to be n>10 43000 . Vinogradov gave the asymptotic representation
count of
R3(n) =#{(p, q, r) : n = p + q + r; p, q, r ∈P} (1.11)
as
R3(n) =Θ(n) n2
2ln 3
ln ln n
1+O
, (1.12)
n
ln n
where Θ is the so-called singular series for the ternary Goldbach problem,
namely
Θ(n) =
p∈P
1+
1
(p − 1) 3
1 −
p|n,p∈P
1
p2
.
− 3p +3