On the Distribution of Twin Primes
Madieyna Diouf

TL;DR
This paper introduces a new sieve method for counting twin primes up to a certain range, providing asymptotic bounds and insights into their distribution, which could lead to progress on prime number conjectures.
Contribution
It develops a novel sieve approach dependent on a parameter, offering a new perspective on the distribution and asymptotic bounds of twin primes.
Findings
Established an asymptotic bound for twin primes less than x
Introduced a fundamental structure for twin prime distribution
Proposed a variant of the prime number theorem
Abstract
We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of , establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than .
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Identities · Advanced Algebra and Geometry
On the distribution of twin primes
Madieyna Diouf
Madieyna Diouf
Arizona State University, Tempe, AZ 85282
(Date: 08/08/2021)
Abstract.
We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of , establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than .
Key words and phrases:
Twin primes, Sieve theory, prime numbers, prime counting function
Key words and phrases:
prime numbers, twin primes, distribution of primes
2010 Mathematics Subject Classification:
11N05
2010 Mathematics Subject Classification:
11N05, , 11N35, 11N36
1. Introduction And Statement of Results
The following statements describe our findings including key quantitative and technical results on a sieve for counting twin primes. A fundamental structure of the distribution of twin primes is obtained by sieve in
[TABLE]
where is the average number of composite integers between the pairs of consecutive primes whose gap exceeds two and its lower member is less than .
This opens possibilities to reach some asymptotic and non-asymptotic bounds of the number of primes and twin primes up to . Consequently, we prove the following variant of the prime number theorem by combining respectively
[TABLE]
The aforementioned results are explicit and unconditional. By using and invoking , one has the main result in Theorem
[TABLE]
Clearly establishes an asymptotic bound of . Our method revolves around a sieve that counts twin primes by enumerating their midpoints.
1.1. Background
Let’s first give a brief history of the twin prime conjecture and outline some recent breakthroughs making dramatic new progress on questions related to the conjecture.
Conjecture 1** (Twin Prime Conjecture).**
There are infinitely many twin primes.
G. H. Hardy and J.E. Littlewood [20] formulated a similar, but stronger twin prime conjecture known as the first Hardy-Littlewood conjecture.
Conjecture 2** (Strong Twin Prime or first Hardy-Littlewood conjecture).**
[TABLE]
In , the Twin Prime conjecture was considered by the French Mathematician Alphonse de Polignac [1], but there has been speculation that it could go back as Euclid and the ancient Greeks over 2000 years ago. Attempts to prove the conjecture have produced partial results that stand on their own merits. Results including Brun’s theorem [4], Mertens’ theorems [10] and the Hardy-Littlewood conjecture [8], along with Chen’s theorem [5].
Currently, the result obtained by sieve on twin primes is an upper bound that states the following. There exists a constant such that for
[TABLE]
The best value of obtained in this direction is slightly below whenever is large enough (see [11], building on the earlier work of [7], [13], [14], [17], [15]).
The original work of the pioneer of modern Sieve Theory Viggo Brun, implied a convergence of the sum of reciprocals of twin primes.
Chen has proved that There are infinitely many primes such that has at most prime factors. Unfortunately, on a march toward the twin prime conjecture, his method faces the parity problem that prevents sieves from giving good estimates.
Bounded gaps between primes is, in recent years, the most used alternative approximation to the Twin Prime Conjecture. In the groundbreaking work of Goldston-Pintz-Yıldırım [22], they proved that given , there are infinitely many pairs of distinct primes such that
[TABLE]
Their method also showed that if the Elliott-Halberstam conjecture is true, then the gap above can be reduced to . Despite many significant results in this direction [12], [17], [18], [19], it is not until that Zhang [2] proved an acceptable alternative of this conjecture, which allowed him to use the GPY method to prove that there are infinitely many bounded gaps between primes. Zhang’s work validates the existence of infinitely many pairs of distinct primes such that
[TABLE]
In less than a year after Zhang’s results, a new variant of the GPY method was discovered independently by Maynard [9] and Tao (unpublished).
A great deal of effort has already been expended on the Twin Prime conjecture. It is widely believed that new ideas may be required to prove the conjecture.
A fair question is, how does the idea presented here differ from those of the previous authors? The improvement comes from a new sieve for finding twin primes. It offers a platform, where with a little bit of effort, one can make a significant contribution to the twin prime conjecture by engineering good bounds of for .
2. A sieve for counting twin primes
2.1. Preliminary Steps
Consider a composite odd integer ; fixed, once for all.
Let be the largest prime not exceeding .
We denote by , the number of pairs of consecutive primes whose gap is two or less. These are the twin primes and the pair .
Say is a sequence of composite integers between two consecutive primes whose gap exceeds two and its lower member is less than .
Set to be the number of these sequences less than . Thus,
, , , …, .
By the definition of , the last sequence denoted by , is expected to be
. But since and we are only interested in the integers , then the last sequence shall end at to be .
Let be the set of sequences , for going from to ,
We define as the set of pairs of consecutive primes whose gap exceeds two and its lower member is less than ,
The notation indicates a pair of consecutive primes and .
Observe that and are different sets with the same cardinality. That is
[TABLE]
Let hold the average number of composites per sequence in . Clearly, this is the same definition of as given in the introduction. Meaning that points to the average number of composites between the pairs of consecutive primes whose gap exceeds two and its lower member is less than .
2.2. Sieving for twin primes
At the presence of the positive integers less than or equal to ,
a) Remove all primes .
b) Remove all elements in the .
The remaining integers are the midpoints of the twin primes , except the positive integer . We wish to count these midpoints since each of them represents a unique pair of twin primes. The positive integer shall count for the midpoint of the first pair which is not a twin prime and its midpoint is not represented in the sieve. Proceedings of steps and are expressed by:
[TABLE]
Where
[TABLE]
The approximation symbol in and in the remainder of this paper, shall be defined as “almost equal to”. It is used above to indicate that the real value of is almost equal to its estimation . In , we state that the number of integers in the union of sequences elements of the set , is approximately equal to the number of sequences in the set that is , multiplied by the average number of integers per sequence denoted by .
With and we get
[TABLE]
c) Given the primes .
Remove the (lower members) of the pairs of consecutive primes whose gap exceeds two.
The remaining must be (the lower members) of the pairs whose gap is two or less; This gives the value of by definition. Step is formulated as
[TABLE]
Since
[TABLE]
then substituting in with gives
[TABLE]
With and , we obtain
[TABLE]
2.3. Quick Example
Say , then
[TABLE]
The value of in this example is a better approximation of than . In addition, the number of twin primes (including the pair ) is . When using estimations obtained by sieve, we have
[TABLE]
The estimation of in frames, a fundamental structure of the distribution of twin primes. It suffices now to find a good bound for .
3. An asymptotic bound of
This section is devoted to proving that . It requires Lemma and together with Theorem , before the result is exhibited in Theorem .
[TABLE]
Let be the real value that satisfies the equation
[TABLE]
It is well known that when as the result of the prime number theorem.
Going forward, the symbol shall be read as “less than or almost equal to”.
Lemma 3.1**.**
[TABLE]
Proof.
By following the steps in the sieve and looking for instead of ,
we obtain by
[TABLE]
In , replace with its expression given in , we get
[TABLE]
By
[TABLE]
With and we have
[TABLE]
where
[TABLE]
By Dusart [16]: If , then
[TABLE]
A combination of and implies that
[TABLE]
With and , one has
[TABLE]
By using to simplify the left inequality in , we get
[TABLE]
So that
[TABLE]
∎
Lemma 3.2**.**
[TABLE]
Proof.
Of Lemma , we deduce via that
[TABLE]
∎
3.1. A suitable variant of the prime number theorem
Theorem 3.3**.**
[TABLE]
Proof.
In , we defined as
[TABLE]
By , we have
[TABLE]
It is clear from and that
[TABLE]
As a result of Lemma , we have
[TABLE]
Dividing by , then taking the limit of the right side shows that
[TABLE]
Similarly, taking the limit of the left side of after dividing by yields
[TABLE]
The inequalities in combined with and imply that
[TABLE]
∎
Theorem 3.4**.**
[TABLE]
Proof.
By the prime number theorem, we have
[TABLE]
It is known from Theorem that
[TABLE]
Statements and imply that
[TABLE]
∎
4. An asymptotic bound of
Since is established, we proceed to prove the following.
Theorem 4.1**.**
[TABLE]
Proof.
By ,
[TABLE]
Recall that in , we set
[TABLE]
where
[TABLE]
With and , we obtain
[TABLE]
By Theorem ,
[TABLE]
This also means that
[TABLE]
Substituting with in yields
[TABLE]
Set
[TABLE]
then
[TABLE]
Because , we have .
The last sentence combined with imply that
[TABLE]
Using statement to replace with in , gives a leading constant of
[TABLE]
With and , we obtain
[TABLE]
∎
5. Remarks
1) The sign in the leading constant in , is believed to be a negative operator. This means that should be
[TABLE]
To validate claim , one must show that
[TABLE]
Unfortunately, we could not provide a proof of to justify Claim . For this reason, we have a sign in our leading constant in , in lieu of a solid minus sign.
2) We shall also note that, there is a possible deviation of our leading constant
, from what the elusive yet enlightening Hardy-Littlewood conjecture predicts; That is a constant value of Nevertheless, the main result in has heretofore been out of reach.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] A. de Polignac. Recherches nouvelles sur les nombres premiers. Comptes Rendus Acad. Sci., (29):397–401, 1849.
- 2[2] Zhang, Yitang. ”Bounded gaps between primes.” Annals of Mathematics (2014): 1121-1174.
- 3[3] D. A. Goldston, J. Pintz, and C. Y. Yıldırım. Primes in tuples. I. Ann. of Math. (2), 170(2):819–862, 2009.
- 4[4] V. Brun, Le crible d’Eratosthéne et le théorème de Goldbach, Skr. Norske Vid.-Akad. Kristiania I, No. 3, 1920.
- 5[5] J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157—176.
- 6[6] E. Bombieri, J. B. Friedlander, and H. Iwaniec. Primes in arithmetic progressions to large moduli. Acta Math., 156(3-4):203–251, 1986.
- 7[7] J. R. Chen. On the Goldbach’s problem and the sieve methods. Sci. Sinica, 21(6):701–739, 1978.
- 8[8] G. H. Hardy and J. E. Littlewood. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Math., 44(1):1–70, 1923.
