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 the previous post we introduced a new kind of summation, the one extended to the divisors of a positive integer number. Summations of this kind are often used in number theory, so it's worth to analyse them in more details. The most interesting case is the one of double summations extended to divisors: we'll see that they can be written in simpler ways, especially when the Möbius function comes into play.
The properties of the divisors of natural numbers which we saw in the previous post let us define a function which is very important in number theory, the Möbius function, indicated by the symbol μ. It's often used in summations in which the variable does not vary between a minimum and a maximum, but it assumes as possible values all and only the positive divisors of a natural number.
In this post we'll illustrate two properties of the divisors of natural numbers, starting from the simplest case, in which we'll take into consideration the numbers which are the product of two distinct prime factors. Let's take for example the number 10, which is the product of 2 and 5. So its unique non-trivial divisors, the ones different from the number itself and 1, are just 2 and 5. You can note that the number of trivial divisors (1 and 10) is equal to the number of non-trivial ones (2 and 5), and also their product is equal: 1 *…
In this post we'll apply the mean value Theorem for integrals in order to transform what we know about the integral of the function V into a knowledge about the values assumed by the function, aiming to prove the sufficient condition for the Prime Number Theorem, that is lim sup |V(u)| = 0. Our calculations will let us introduce a new function R that represents an absolute error, contrary to the functions V and W that represent relative errors.
Background Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, the Russian Empire was always one step behind, with an industry that was little more than artisanship, a feudal economy, and a public education that wasn't even worthy of the name. But the new Tsar Peter the Great, after touring Europe in search of allies against the Turks who threatened the southern borders, was able to see this abysmal gap with his own eyes, and decided that the time had come for a turning point. He immediately started a great reform…