Sieve theory

Sieve theory is a mathematical theory, which falls within the general scope of Number Theory, born to answer the following question: given a set of natural numbers $A$ and a finite set of prime numbers $P$ , how many elements remain in $A$ after eliminating all the multiples of the prime numbers present in $P$ ? If $A$ and $P$ are fixed, the answer can be obtained quite simply, because it is sufficient to follow a series of steps, such as happens in the sieve of Eratosthenes; since the steps are known, the procedure can also be easily automated with a simple computer program. If instead we wanted to know how the answer varies as a function of $A$ and $P$ , the program would not be of much help, because it would only provide answers for single combinations of parameters, while then it would be necessary to find a different mechanism to generalize the results obtained, so that it is possible to predict the result even for combinations of $A$ and $P$ that have not been tested. Sieve theory has precisely this purpose: first of all it defines a sieve not as a procedure, but as a function that, applied to the sets $A$ and $P$ , returns the number of elements of $A$ that are not divisible by any element of $P$ ; then, since this is almost always a mathematical function of a certain complexity, it tries to approximate it with a simpler function. In the following paths we’ll delve into some techniques that allow us to perform this approximation and that are used in one of the most important theorems of sieve theory: Chen’s Theorem.

The required prerequisites are, as in all our site, a good knowledge of school mathematics and the basics of real analysis in one variable (derivatives, integrals, continuity, etc.). Other prerequisites will be indicated at the beginning of each article; for example, many articles presuppose a certain knowledge of asymptotic analysis, which we have covered in the Complementary material path.

The source we started from is a master’s thesis publicly available on the website of the University of Barcelona. We have completely reworked the work, enriching it with our observations and explanations, breaking complex passages into a series of simpler passages and providing references for the notions of number theory used.

Path	Number of posts
An overestimate of the number of Goldbach pairs This path studies the set of Goldbach couples for an even number $N \gt 2$ , that is the set of the couples made up of prime numbers the sum of which is $N$ . While Goldbach’s conjecture states that such set always contains at least one element for every $N$ , in this path we’ll study the same couples from an opposite perspective: we’ll try to establish how many they can be at most. Indicating with $\mathfrak{R}(N)$ the number of Goldbach couples for $N$ , we’ll prove that: $\mathfrak{R}(N) \ll \prod_{p \mid N} \left(1 + \frac{1}{p}\right) \frac{N}{\log^2 N}$ where the notation $\ll$ is equivalent to the big Oh notation (simplifying, it’s a sort of approximate overestimation, which allows that, as $N$ increases, the right function can become $C$ times greater than the actual value, where $C$ is a positive constant). Though this result by itself is not helpful for proving the Conjecture, the interesting aspect is that it’s the only known theorem which somehow tries to approximate the number of Goldbach couples, while other theorems study similar but not identical sets (terns of prime numbers in the case of weak Goldbach’s Conjecture, couples made up of a prime and a semiprime in the case of Chen’s Theorem). Will some our reader be successful at inverting the perspective, transforming the $\leq$ into a $\geq$ , paving the way for a possible proof of Goldbach’s Conjecture?	8
Chen’s Theorem The goal of this path is to prove one of the theorems most similar to Goldbach’s conjecture, Chen’s Theorem, the statement of which can be expressed as follows: Every sufficiently large even number is the sum of either two primes, or a prime and a semiprime (that is, a product of two primes). This Theorem was proved for the first time by the Chinese mathematician Chen Jingrun in 1966 (subsequently the proof was refined both by himself and by other mathematicians). The proof is based on a mathematical theory called sieve theory, developed a few years earlier but with very ancient origins, which can be traced back to the mathematician of ancient Greece Eratosthenes of Cyrene, inventor of the homonymous sieve, the first ever. Clearly, since then sieve theory has evolved a lot, but it has always remained an “elementary” theory, i.e. one not based on complex analysis.	9 (incomplete)

Path

Number of posts

An overestimate of the number of Goldbach pairs

This path studies the set of Goldbach couples for an even number $N \gt 2$ , that is the set of the couples made up of prime numbers the sum of which is $N$ . While Goldbach’s conjecture states that such set always contains at least one element for every $N$ , in this path we’ll study the same couples from an opposite perspective: we’ll try to establish how many they can be at most.
Indicating with $\mathfrak{R}(N)$ the number of Goldbach couples for $N$ , we’ll prove that:

\mathfrak{R}(N) \ll \prod_{p \mid N} \left(1 + \frac{1}{p}\right) \frac{N}{\log^2 N}

where the notation $\ll$ is equivalent to the big Oh notation (simplifying, it’s a sort of approximate overestimation, which allows that, as $N$ increases, the right function can become $C$ times greater than the actual value, where $C$ is a positive constant).
Though this result by itself is not helpful for proving the Conjecture, the interesting aspect is that it’s the only known theorem which somehow tries to approximate the number of Goldbach couples, while other theorems study similar but not identical sets (terns of prime numbers in the case of weak Goldbach’s Conjecture, couples made up of a prime and a semiprime in the case of Chen’s Theorem). Will some our reader be successful at inverting the perspective, transforming the $\leq$ into a $\geq$ , paving the way for a possible proof of Goldbach’s Conjecture?

Chen’s Theorem

The goal of this path is to prove one of the theorems most similar to Goldbach’s conjecture, Chen’s Theorem, the statement of which can be expressed as follows:

Every sufficiently large even number is the sum of either two primes, or a prime and a semiprime (that is, a product of two primes).

This Theorem was proved for the first time by the Chinese mathematician Chen Jingrun in 1966 (subsequently the proof was refined both by himself and by other mathematicians). The proof is based on a mathematical theory called sieve theory, developed a few years earlier but with very ancient origins, which can be traced back to the mathematician of ancient Greece Eratosthenes of Cyrene, inventor of the homonymous sieve, the first ever. Clearly, since then sieve theory has evolved a lot, but it has always remained an “elementary” theory, i.e. one not based on complex analysis.

9 (incomplete)