Just a curiosity: What is the possible relation between the twin prime conjecture and the Goldbach's conjecture stating that every even integer greater than $2$ can be expressed as the sum of two primes. I have no idea, but I want to see a relation between them.
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2$\begingroup$ Why do you think there is one? They are quite different topics. Nowadays, by Maynard, Tao and others we know that the limsup of the prime gap is $\leq 246$, but the Chen's result that every even number big enough can be written as the sum of a prime and at most a semiprime has not been improved since a long time, so the twin prime conjecture looks to be a weaker statement than Goldbach's conjecture. $\endgroup$ – Jack D'Aurizio Oct 29 '15 at 16:29
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First of all, excuse my bad english, I am french. Second I must tell that my vision of those mathematical problems is not a "protocolary" (or "academic") one because I am a computer scientist. I worked a lot on binary Goldbach's conjecture (from 2005 to 2014) and on twin primes conjecture also (but not so intensively). I wrote in 2013 a small note (2 pages in french) explaining that you can see CG as a relative problem (in the sense relative to the even number you want to decompose in a sum of two primes) while twin primes is a sort of "absolute problem" corresponding to this absolute one. The web address for this note is : http://denise.vella.chemla.free.fr/invariante.pdf Perhaps it will interest you. Sincerely yours, Denise Vella-Chemla
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As illustrated by one unified function which zeros show the distribution of relative prime, twin prime, prime pair of distance 2n (twin prime is special case of n=1, and prime is special case of n=0), and Goldbach sums of 2n, there indeed has relationship between these problems. Here's a summary and you can see more details from my research notes: https://fredyangblog.files.wordpress.com/2016/04/fourierseriesofprimes-rev1-3.pdf
I also created a live chart to demonstrate this: https://www.desmos.com/calculator/4a9i0ejeyk
Let $p_i$ be the $i^{th}$ prime, define $$ P(p_i,n,x)=\sum_{p\le{p_i}}\frac{c_p}{p}\left(1+2\sum_{k=1}^{p-1}(1-\frac{k}{p})\cos\frac{2kn\pi}{p}\cos\frac{2k\pi}{p}x\right), c_p = \begin{cases} 1, \text{ when $p \mid 2n$} \\ 2, \text{ when $p \nmid 2n$} \end{cases} $$ which zeros show $$ \begin{cases} \text{When $n=0$: prime distribution} \\ \text{When $n=1$: twin prime distribution as $(x-1,x+1)$} \\ \text{When $n>1$ and $0\le{x}<n$: distribution of Goldbach sums as $(n-x, n+x)$} \\ \text{When $n\ge{1}$ and $x>n$: distribution of prime pairs of distance of $2n$ as $(x-n, x+n)$} \end{cases} $$
Additionally, when $n=0$, for each integer $x$, it could show the number of prime divisors of $x$ that $\le p_i$.
The unified formula to calculate number of zeros $L$ on $[0,p_i\#)$ is, for all $3\le{p}\le{p_i}$, \begin{equation} L=\prod_{p|n}(p-1)\prod_{p \nmid n}(p-2) \end{equation}
For prime series, $n=0$ so $L=\prod(p-1)$; For twin prime series, $n=1$ so $L=\prod(p-2)$.
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Between the twin primes conjecture and the Goldbach conjecture there is the most direct connection. A generalization of the twin primes conjecture is the assumption that for any even $ g $ there are infinitely many pairs of primes with difference $ g $. The Goldbach conjecture states that any $ g> 2 $ is the sum of two primes. Both problems can be solved by constructing a double sieve. A short version of the construction of a double sieve (only for $ p_ {n + 1} - p_n = 2 $) you can see here . The complete version of the solution of the problems of representing all even numbers as the difference between two primes and the sum of two primes can be found here .
