Prime Numbers

20 Chapter 1 PRIMES!
It is not hard to see that Θ(n) for odd n is bounded below by a positive
constant. This singular series can be given interesting alternative forms (see
Exercise 1.68). Vinogradov’s effort is an example of analytic number theory
par excellence (see Section 1.4.4 for a very brief overview of the core ideas).
[Zinoviev 1997] shows that if one assumes the extended Riemann
hypothesis (ERH) (Conjecture 1.4.2), then the ternary Goldbach conjecture
holds for all odd n>10 20 . Further, [Saouter 1998] “bootstrapped” the then
current bound of 4 · 10 11 for the binary Goldbach problem to show that the
ternary Goldbach conjecture holds unconditionally for all odd numbers up
to 10 20 . Thus, with the Zinoviev theorem, the ternary Goldbach problem is
completely solved under the assumption of the ERH.
It follows from the Vinogradov theorem that there is a number k such
that every integer starting with 2 is a sum of k or fewer primes. This corollary
was actually proved earlier by G. Shnirel’man in a completely different
way. Shnirel’man used the Brun sieve method to show that the set of even
numbers representable as a sum of two primes contains a subset with positive
asymptotic density (this predated the results that almost all even numbers
were so representable), and using just this fact was able to prove there is such
anumberk. (See Exercise 1.44 for a tour of one proof method.) The least
number k0 such that every number starting with 2 is a sum of k0 or fewer
primes is now known as the Shnirel’man constant. If Goldbach’s conjecture is
true, then k0 = 3. Since we now know that the ternary Goldbach conjecture
is true, conditionally on the ERH, it follows that on this condition, k0 ≤ 4.
The best unconditional estimate is due to O. Ramaré who showed that k0 ≤ 7
[Ramaré 1995]. Ramaré’s proof used a great deal of computational analytic
number theory, some of it joint with R. Rumely.
1.2.4 The convexity question
One spawning ground for curiosities about the primes is the theoretical issue
of their density, either in special regions or under special constraints. Are there
regions of integers in which primes are especially dense? Or especially sparse?
Amusing dilemmas sometimes surface, such as the following one. There is an
old conjecture of Hardy and Littlewood on the “convexity” of the distribution
of primes:
Conjecture 1.2.3. If x ≥ y ≥ 2, then π(x + y) ≤ π(x)+π(y).
On the face of it, this conjecture seems reasonable: After all, since the primes
tend to thin out, there ought to be fewer primes in the interval [x, x + y] than
in [0,y]. But amazingly, Conjecture 1.2.3 is known to be incompatible with
the prime k-tuples Conjecture 1.2.1 [Hensley and Richards 1973].
So, which conjecture is true? Maybe neither is, but the current thinking is
that the Hardy–Littlewood convexity Conjecture 1.2.3 is false, while the prime
k-tuples conjecture is true. It would seem fairly easy to actually prove that the
convexity conjecture is false; you just need to come up with numerical values of
x and y where π(x+y),π(x),π(y) can be computed and π(x+y) >π(x)+π(y).