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 this post we'll complete the proof of the prime number Theorem, applying the fundamental ideas described in the previous post. We'll split this last part of the proof into five parts: Our starting point will be a simple overestimation of the integral of |V| in any interval, obtained by applying Property A.18; We'll improve the result of the previous point supposing that within the interval the function V has at least one zero; We'll improve the result of the first point also for the intervals in which the function V has no zeroes; We'll compute how long the length…
In this post we’ll see what are the basic ideas of the second part of the prime number Theorem proof. Up to now we proved that α ≤ β′, where α and β′ are two important constants connected with the function V. As we saw, if we proved that α = 0 we would have proved the prime number Theorem. In order to prove that, a first idea may be proving that che β′ = 0, because it would follow that α ≤ β′ = 0, hence α = 0. In this post we’ll try to prove that β′ =…
With this post we'll conclude the main part of the Prime Number Theorem proof, which is based on the relationship between the constants α and β which have been introduced in the post The integral mean value and the absolute error function R. By Lemma N.7, in order to prove the Prime Number Theorem it's sufficient to prove that α = 0: in this post we'll approach this result.
Looking back to the path travelled so far, we can identify a turning point: it was when we introduced Hypothesis N.1, which consists in an inequality with an integral inside it. At that point, in order to simplify, we decided to replace the integral with a summation and, since then, we have always worked with summations, up to the proof of Selberg's Theorem and some consequences of it. Now it's time to return to the integral form, for resuming the argument which brought us to Hypothesis N.1, trying to prove it.
After the digression about the Möbius function of the last three posts, let's come back to the proof of the Prime Number Theorem. In this post we'll see one of the most important parts of the proof, which consists in the application of Selberg's Theorem. We'll try to understand, with as few technicisms as possible, why this Theorem has a so important role in the proof of the Prime Number Theorem.