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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Little late to the conversation here but there is a connection I have thought of a couple of years back.
basically if you express Goldbach Conjecture as $~2n=P_1 + P_2~$ then divide both sides by $~2~$ so $~n=\frac{1}{2}(P_1 +P_2)~$ this is the average of $~2~$ primes
so with this an equivalent statement of the Conjecture is that every number $~\ge 2 ~$is the average of $~2~$ primes
then you can think about what averaging $~2~$ numbers mean, and that is that the number is equal-distance from both numbers
after that you can group numbers based on how far they are from the primes that they are the average of, so $~P_1 - n = k~$ (where $~P_1~$ is the larger of $~P_1 ~$and $~P_2~$)
then the group of numbers that are $~k=1~$ represent twin primes, and if that subset of the natural numbers is infinite then that is the twin prime conjecture.
