The goal
This path studies the set of Goldbach couples for an even number N \gt 2, that is the set of the couples made up of prime numbers the sum of which is N. While Goldbach’s conjecture states that such set always contains at least one element for every N, in this path we’ll study the same couples from an opposite perspective: we’ll try to establish how many they can be at most.
Indicating with \mathfrak{R}(N) the number of Goldbach couples for N, we’ll prove that:
where the notation \ll is equivalent to the big Oh notation (simplifying, it’s a sort of approximate overestimation, which allows that, as N increases, the right function can become C times greater than the actual value, where C is a positive constant).
Though this result by itself is not helpful for proving the Conjecture, the interesting aspect is that it’s the only known theorem which somehow tries to approximate the number of Goldbach couples, while other theorems study similar but not identical sets (terns of prime numbers in the case of weak Goldbach’s Conjecture, couples made up of a prime and a semiprime in the case of Chen’s Theorem). Will some our reader be successful at inverting the perspective, transforming the \leq into a \geq, paving the way for a possible proof of Goldbach’s Conjecture?
The path
Before arriving to the actual proof, we’ll introduce sieve theory, starting from the definition of the oldest and most famous sieve, the one of Eratosthenes. Next we’ll study a more modern sieve technique, Selberg’s sieve, which will let us achieve the final result with the help of two auxiliary Lemmas.
General introduction to sieve theory paths