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.
In number theory, to understand some proofs, a certain familiarity with the concept of "numerical series" is necessary. In this article, after briefly introducing this concept, we'll focus on some series used in number theory, which are not always studied at school or university; we'll see in particular some techniques which are useful in practice to operate with such series. This article does not claim to treat numerical series in a rigorous nor exhaustive way, but it can be seen as a complementary study for those who have already studied series, and as a starting point for further study for…
In this article we'll discuss arithmetic functions, that are functions defined on positive integers; the value of such functions, however, may not always be an integer. Some arithmetic functions are well-known and recurrent in number theory, and we have introduced them in our articles where appropriate. Here we'll summarize their definitions and then we'll see some of their properties that can potentially be of common use, also with regard to the techniques used in some proofs.
We have seen that, thanks to Selberg's sieve, an estimate of a sieve function can be calculated; this estimate is made up of a term T which represents the approximation of the function, and a term R which represents the error. In the previous article we calculated the λ parameters that make the T part as low as possible. Now all that remains is to replace these parameter values in the R error function; this will be the purpose of the present article.
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.