## Why don’t algorithmic approaches work well in sieve theory? (part II)

In the previous article we calculated the sieve function of Erathostenes' sieve starting from the algorithm, obtaining a formula with…

## Chen’s Theorem: statement and introduction to the proof

Chen's Theorem is one of the closest theorems most similar to Goldbach's Conjecture known so far. It is the work…

## Study about the existence of complementary space pairs based on second order dashed lines

Prerequisites: Dashed line theory definitions and symbols Our proof strategies Proof strategy based on dashes Characterization of spaces The aim…

## The prime number Theorem: end of proof

In this post we'll complete the proof of the prime number Theorem, applying the fundamental ideas described in the previous…

## The second part of the prime number Theorem proof: the basic ideas

In this post we’ll see what are the basic ideas of the second part of the prime number Theorem proof.…

## End of the first part of the proof: the relationship between α and β’

With this post we'll conclude the main part of the Prime Number Theorem proof, which is based on the relationship…

## A consequence of Selberg’s Theorem, in integral form

Looking back to the path travelled so far, we can identify a turning point: it was when we introduced Hypothesis…

## Some important summations

When studying number theory, you'll soon realize that some familiarity with certain formalisms is required. In particular, some kinds of…

## Selberg’s Theorem: proof and application

After the digression about the Möbius function of the last three posts, let's come back to the proof of the…

## Factorizer

This page allows performing the decomposition of a number into its prime factors, and computing the value of some arithmetic…

## Two lemmas with the Möbius function and the logarithm

In number theory, many proofs are "technical", i.e. they consist mainly in algebrical passages, by means of which an initial…

## The Möbius inversion formula

In the post Some important summations we introduced the summations extended to couples of variables the product of which divides…

## The Möbius function and its connection with the function Λ

The properties of the divisors of natural numbers which we saw in the previous post let us define a function…

## Two properties of the divisors of natural numbers

In this post we'll illustrate two properties of the divisors of natural numbers, starting from the simplest case, in which…

## The integral mean value and the absolute error function R

In this post we'll apply the mean value Theorem for integrals in order to transform what we know about the…

## The functions W and V

The general idea of the Prime Number Theorem proof consists in starting from the proof of Chebyshev's Theorem (strong version),…

## The Prime Number Theorem: history and statement

Looking at a prime numbers table, it's very simple to notice how their distribution seems to escape any regularity; instead…

## Chebyshev’s Theorem (strong version)

In this post we'll revisit Chebyshev's Theorem, according to which the function π(x), that counts the number of prime numbers…

## The factorial function and the Λ* function

Almost certainly you already know the factorial function, indicated by x!, which is read as "x factorial" and for an…

## 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…

## Number theory

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer…

## Goldbach’s Conjecture

The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in…

## 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 +…

## 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…

## Chebyshev’s Theorem (weak version)

With this post we begin an analytical study of the function pi(x), that returns the number of primes less than…

## 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…

## The least common multiple of the first positive integers

We know that a way to compute the least common multiple between two or more integer numbers is based on…

## Bertrand’s postulate

The goal of this post is to prove the Bertrand's postulate, proposed in 1845 by the French mathematician Joseph Louis…

## The product of the first prime numbers: an overestimation

A way to start investigating the sequence of prime numbers is to consider, starting from the beginning, portions of increasing…

## Binomial estimates

Binomial coefficients are important for studying prime numbers. In this post we see in particular how to estimate, both upwards…

## The definition of prime number

We'll start our study of prime numbers explaining the definition of prime number. It's commonly known that a prime number…