## Selberg’s sieve: study of the error term

We have seen that, thanks to Selberg's sieve, an estimate of a sieve function can be calculated; this estimate is…

Multa non quia difficilia sunt non audemus sed quia non audemus sunt difficilia

We have seen that, thanks to Selberg's sieve, an estimate of a sieve function can be calculated; this estimate is…

As we have seen, some parameters λd appear in the Selberg's sieve formula, which are used both in the estimation…

In this article we'll begin to get to the heart of sieve theory, analyzing in detail the so-called "Selberg's sieve".…

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

In the previous article we examined in detail the sieve of Eratosthenes, both at an algorithmic level and as a…

In the previous article we saw that the proof of Chen's Theorem is based on sieve theory. But what is…

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

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

Prerequisite: Proof strategy based on spaces One of the strategies that we have developed to try to prove Goldbach’s Conjecture…

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

Prerequisite: Our proof strategies: an overview The proof strategy set out here starts from one of the assumptions of the…

Prerequisite: Our proof strategies: an overview The proof strategy which will be exposed here has the goal of proving Hypothesis…

Prerequisites: Our proof strategies Factorization dashed lines The final aim of the proof strategies that we are carrying out, as…

Prerequisite: Goldbach’s conjecture As already indicated, our ultimate goal is to use dashed line theory to prove Goldbach’s conjecture. Dashed…

The following map presents an overview of all paths related to proof strategies: the items listed below correspond to paths,…

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

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

The Maximum space distance calculator is a program which we developed for computing the value of the maximum distance between…

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

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

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

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

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

In number theory, many proofs are "technical", i.e. they consist 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 post Some important summations we introduced the summations extended to couples of variables the product of which divides…

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…

The function [latex]\mathrm{t\_value}[/latex], by definition, indicates which column of a dashed line a dash belongs to. For this reason, in…

One of the still open problems of dashed line theory is, given a linear dashed line [latex]T = (p_1, p_2,…

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

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…

Molte cose non è perché sono difficili che non osiamo farle,

ma è perché non osiamo farle che sono difficili

Many times, it is not because things are difficult that we do not dare,

but it is because we do not dare that things are difficult

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