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.

11. The sum of inverses of the first positive integers

In this post we'll analyze the sum of the first positive integers: 1 + 1/2 + 1/3 + 1/4 + ... . The theory of numerical series tells us that it's a diverging series. This means that we can fix any positive number M and be certain that, no matter how big it is, the so called partial sums 1, 1 + 1/2, 1 + 1/2 + 1/3, ..., sooner or later we'll exceed it. But what if we want to have an idea of how big these partial sums are, as the number of terms increases?
CONTINUE

12. From integer numbers to real numbers – second part

In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function defined on integer numbers, by means of an appropriate integral of a monotonous extension of it. We'll apply this technique to a logarithmic function, obtaining a lemma that will turn out to be useful in the future.
CONTINUE

13. The factorial function and the Lambda* function

Almost certainly you already know the factorial function, indicated by x!, which is read as "x factorial" and for an integer x > 0 it consists in the product of all positive integers up to x. In this post we'll study this function, that often occurs in number theory, from two different points of view: first we'll study its order of magnitude; then we'll decompose the product x! in such a way to highlight the prime factors which it's made up of.
CONTINUE

14. The limit inferior and the limit superior of a sequence

In this post we'll see two concepts of mathematical analysis which will be useful in number theory: the limit inferior (lim inf) and the limit superior (lim sup) of a sequence. We'll see that they can be defined starting from the study of the intervals that contain all, or almost all, the terms of the sequence.
CONTINUE

11. The sum of inverses of the first positive integers

In this post we'll analyze the sum of the first positive integers: 1 + 1/2 + 1/3 + 1/4 + ... . The theory of numerical series tells us that it's a diverging series. This means that we can fix any positive number M and be certain that, no matter how big it is, the so called partial sums 1, 1 + 1/2, 1 + 1/2 + 1/3, ..., sooner or later we'll exceed it. But what if we want to have an idea of how big these partial sums are, as the number of terms increases?
CONTINUE

13. The factorial function and the Lambda* function

Almost certainly you already know the factorial function, indicated by x!, which is read as "x factorial" and for an integer x > 0 it consists in the product of all positive integers up to x. In this post we'll study this function, that often occurs in number theory, from two different points of view: first we'll study its order of magnitude; then we'll decompose the product x! in such a way to highlight the prime factors which it's made up of.
CONTINUE

14. The limit inferior and the limit superior of a sequence

In this post we'll see two concepts of mathematical analysis which will be useful in number theory: the limit inferior (lim inf) and the limit superior (lim sup) of a sequence. We'll see that they can be defined starting from the study of the intervals that contain all, or almost all, the terms of the sequence.
CONTINUE