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…
Category: Number theory
Other conjectures related to sums of primes
Over the centuries, various scholars have attempted to relate odd and even numbers with sums involving prime numbers. Some of…
Summations extended to the divisors of a natural number and Möbius inversion formula
In the previous post we introduced a new kind of summation, the one extended to the divisors of a positive…
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…
Bidimensional sieve of Eratosthenes
This page allows viewing a “bidimensional” version of the sieve of Eratosthenes applied to a given number. Differently from its…
Goldbach pairs viewer
This page allows viewing all Goldbach pairs in which an even number can be decomposed. The search can be done…
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…
Origins of the two Goldbach’s conjectures
Background Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, 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 Lambda* 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…
Calculation of t_value for dashed lines of arbitrary order
Prerequisites: Dashed line theory definitions and symbols Computation of a dash value in a linear third order dashed line First…
Proof strategy based on factorization
Prerequisites: Our proof strategies Factorization dashed lines Properties of the spaces of the factorization dashed lines can be used to…
Study of 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 purpose…
Upper bound for maximum distance between consecutive spaces
Prerequisite: Characterization of spaces General aspects One of the still open problems of dashed line theory is, given a linear…
Proof strategy based on spaces
Prerequisite: Our proof strategies The proof strategy shown here is aimed at proving the Hypothesis H.1 (Hypothesis of existence of…
Our proof strategies: an overview
Prerequisites: Goldbach’s conjecture From prime numbers to dashed lines Periodicity and symmetry in linear dashed lines As already stated, our…
Proof strategy based on dashes
Prerequisite: Our proof strategies The proof strategy which will be exposed here starts from one of the premises of Hypothesis…
Number theory
Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer…
Our proof strategies
Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to…
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…
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…
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…
Some important results
The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This…
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