## Two lemmas with the Möbius function and the logarithm

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

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

Over the centuries, various scholars have attempted to relate odd and even numbers with sums involving prime numbers. Some of…

In the previous post we introduced a new kind of summation, the one extended to the divisors of a positive…

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

This page allows viewing a “bidimensional” version of the sieve of Eratosthenes applied to a given number. Differently from its…

This page allows viewing all Goldbach pairs in which an even number can be decomposed. The search can be done…

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

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

Background Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, the…

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

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

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

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

In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function…

Prerequisites: Dashed line theory definitions and symbols Computation of a dash value in a linear third order dashed line First…

Prerequisites: Our proof strategies Factorization dashed lines Properties of the spaces of the factorization dashed lines can be used to…

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

Prerequisite: Characterization of spaces General aspects One of the still open problems of dashed line theory is, given a linear…

Prerequisite: Our proof strategies The proof strategy shown here is aimed at proving the Hypothesis H.1 (Hypothesis of existence of…

Prerequisites: Goldbach’s conjecture From prime numbers to dashed lines Periodicity and symmetry in linear dashed lines As already stated, our…

Prerequisite: Our proof strategies The proof strategy which will be exposed here starts from one of the premises of Hypothesis…

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

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to…

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

In this post we'll analyze the sum of the first positive integers: 1 + 1/2 + 1/3 + 1/4 +…

So far we defined and studied only functions defined on integer numbers, the values of which can be integer or…

The problem we establish in this post is to compute the area of a bar chart. Of course the area…

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

We saw that the product of the first prime numbers can be overestimated by a function of exponential kind with…

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

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

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

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

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

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This…

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