The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:
Every even number greater than 2 can be expressed as a sum of two prime numbers.
In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.
As we have seen, some parameters λd appear in the Selberg's sieve formula, which are used both in the estimation of the sieve function and in the related error term. In this article we'll calculate these parameters in order to obtain the lowest possible estimate.
In this article we'll begin to get to the heart of sieve theory, analyzing in detail the so-called "Selberg's sieve". It is not, in reality, a new sieve (comparable for example to that of Eratosthenes); the sieve (understood as a function) is always the general one, Selberg obtained an overestimation of it. In summary, we can say that what is called "Selberg sieve" is actually a way of overestimate a sieve function with a function simple to calculate, composed of two terms: an estimate and an error term.
In the previous article we calculated the sieve function of Erathostenes' sieve starting from the algorithm, obtaining a formula with a number of terms that grows exponentially with respect to the number of primes used, therefore too complex to be used in practice. But ultimately it is not necessary to calculate a sieve function exactly: to be able to study it (for example to understand its asymptotic behavior) it would be sufficient to find a good approximation. Therefore, if we were content to approximate the sieve function, in the case of Eratosthenes' sieve, would we obtain a less complicated formula?…
In the previous article we examined in detail the sieve of Eratosthenes, both at an algorithmic level and as a sieve function. We have seen that there is a strong connection between the two formulations, as the sieve function calculates how many elements remain in the set {2, 3, 4, ..., N} after having applied the algorithm. But so, to calculate the sieve function, why not start from the study of the algorithm from which it originates? In this article and the next one we'll see that this idea does not work in practice, and we'll understand why.
In the previous article we saw that the proof of Chen's Theorem is based on sieve theory. But what is a sieve? In this article we'll explain it based on the most famous example of a sieve, that of Eratosthenes. It allows you to answer a simple question: given an integer N ≥ 2, what are the prime numbers less than or equal to N?