Number theory

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum. Indeed number theory is considered a very specialistic subject which can be taken under consideration, as an example, for some university exams selected by the student, or a doctorate, for the most enthusiastic people. But it’s not necessary going this far for undertaking a serious study of number theory. Many notions are within reach even of a first year student of a scientifical degree course, since they require just a good knowledge of school level maths and of real analysis in one variable.

We started from these considerations for conceiving this site section, with the aim of allowing approach to number theory as many people as possible, giving at the same time useful elements to who wants to test himself or herself with the proof of the conjecture.
For making our writings more interesting, we focused the itinerary towards a goal, and as a goal we chose one of the most important theorems about prime numbers, the so called Prime number theorem. A bit like in a detective story - in which several evidences are collected and, thanks to them, the various aspects of the story get more and more clear, up to identifying the culprit - in the way towards the proof of the theorem, we'll introduce more and more notions about number theory, and prove several intermediate results, getting more and more close to the final result.

But what does the prime number theorem state? The statement can be expressed by the following formula:

\pi(x) \sim \frac{x}{\log x}

where \pi(x) is the number of primes less than or equal to x, and \log x is the natural logarithm of x. The symbol \sim is an asymptotic equivalence whose meaning will be clear in one of the next posts but, simplifying, we can read the formula by saying that there are about \frac{x}{\log x} prime numbers less than or equal to x, and this estimation becomes more and more accurate as x increases. We encourage you to verify it with different values of x!

The theorem history is interesting by itself. In fact, for a lot of time the only known proofs were based on complex analysis and other quite advanced notions; moreover matematicians thought that complex analysis was somehow "necessary" for the proof. Instead in 1949 Paul Erdős and Atle Selberg independently conceived an "elementary" proof, that is a proof based only on real analysis, and that inspired astonishment in the mathematical community.
The proof we'll present is just the one by Erdős and Selberg. The source we started from is the text "An introduction to the theory of numbers" by G. H. Hardy and E. M. Write. In particular, the proof is contained in chapter XXII, in turn based on the preliminar notions introduced in chapters I and II. They are about 40 pages in total, but very dense. We are reworking this material, adding new original stuff: we introduced some examples, highlighted the key ideas and techniques, and added some details which were not explicit in the original text; we'll also try to enrich the dissertation with various fun facts.

6. The product of the first prime numbers: an underestimation

We saw that the product of the first prime numbers can be overestimated by a function of exponential kind with base 2 and a first degree polynomial as exponent. In this post we see how the same quantity can be underestimated.
CONTINUE

7. Elements of asymptotic analysis

A good part of number theory may be studied by analyzing concrete cases, i.e. starting just from numerical calculations. However, verifying the general truth of what observed on numerical basis cannot be left out of consideration. Sometimes this generalization is immediate, other times it requires the tools of a part of mathematical analysis, called asymptotic analysis. Given the importance of this subject for our purposes, we recall some essential elements.
CONTINUE

8. Chebyshev’s Theorem

With this post we begin an analytical study of the function pi(x), that returns the number of primes less than or equal to x. In particular, we'll compute the order of magnitude of this function. We proceed step by step, studying the order of magnitude of psi(x) and of theta(x) first, recalling that the two functions are strictly connected; then we'll see how their order of magnitude are connected with that of pi(x).
CONTINUE
Figura 6: metodo alternativo per il calcolo dell'area dell'istogramma della Figura 2, secondo passo

9. The lemma of bar chart area

The problem we establish in this post is to compute the area of a bar chart. Of course the area is given by the sum of the areas of the individual rectangles making up the chart, and the area of each rectangle is given by the product of its basis for its height. But we'll see another method, rather creative, for computing the same area. At the end we'll obtain a lemma that will turn useful in several occasions, starting from the next post.
CONTINUE

10. From integer numbers to real numbers

So far we defined and studied only functions defined on integer numbers, the values of which can be integer or real. You could think it's obvious, since our goal is studying integer numbers, in particular prime numbers. However, as often happens in Mathematics, in order to study a certain class of objects it often turns out to be useful to take inspiration from a wider class, by means of an abstraction process. Thus, in order to study integer numbers, we'll see that it's useful to take inspiration from real numbers.
CONTINUE

6. The product of the first prime numbers: an underestimation

We saw that the product of the first prime numbers can be overestimated by a function of exponential kind with base 2 and a first degree polynomial as exponent. In this post we see how the same quantity can be underestimated.
CONTINUE

7. Elements of asymptotic analysis

A good part of number theory may be studied by analyzing concrete cases, i.e. starting just from numerical calculations. However, verifying the general truth of what observed on numerical basis cannot be left out of consideration. Sometimes this generalization is immediate, other times it requires the tools of a part of mathematical analysis, called asymptotic analysis. Given the importance of this subject for our purposes, we recall some essential elements.
CONTINUE

8. Chebyshev’s Theorem

With this post we begin an analytical study of the function pi(x), that returns the number of primes less than or equal to x. In particular, we'll compute the order of magnitude of this function. We proceed step by step, studying the order of magnitude of psi(x) and of theta(x) first, recalling that the two functions are strictly connected; then we'll see how their order of magnitude are connected with that of pi(x).
CONTINUE
Figura 6: metodo alternativo per il calcolo dell'area dell'istogramma della Figura 2, secondo passo

9. The lemma of bar chart area

The problem we establish in this post is to compute the area of a bar chart. Of course the area is given by the sum of the areas of the individual rectangles making up the chart, and the area of each rectangle is given by the product of its basis for its height. But we'll see another method, rather creative, for computing the same area. At the end we'll obtain a lemma that will turn useful in several occasions, starting from the next post.
CONTINUE

10. From integer numbers to real numbers

So far we defined and studied only functions defined on integer numbers, the values of which can be integer or real. You could think it's obvious, since our goal is studying integer numbers, in particular prime numbers. However, as often happens in Mathematics, in order to study a certain class of objects it often turns out to be useful to take inspiration from a wider class, by means of an abstraction process. Thus, in order to study integer numbers, we'll see that it's useful to take inspiration from real numbers.
CONTINUE