# Twin Primes Conjecture and related problems

I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.

Let $$S_f$$ be a random set of positive integers defined as follows: the positive integer $$n$$ belongs to $$S_f$$ with probability $$f(n)$$. In other words, for each integer $$n$$, generate a uniform deviate $$U_n$$ on $$[0, 1]$$. If and only if $$U_n < f(n)$$ then add $$n$$ to $$S_f$$. The deviates $$U_n$$ are independent. Now consider $$f(n) = \frac{1}{\log n}$$ if $$n > 2$$, and $$f(1)= f(2) = 0$$. With this particular choice of $$f$$, the set $$S_f$$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):

• $$M(n)$$ is the number of elements in $$S_f$$ that are smaller than $$n$$.

• $$T(n)$$ is the number of twin pairs $$(k, k+1)$$ with both $$k, k+1 \in S_f$$ and $$k \leq n$$.

The twins in $$S_f$$ play the same role as twin primes. It is is easy to prove that $$M(n)$$ (its expectation) is asymptotically equal to the logarithm integral $$\mbox{Li}(n)$$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $$S_f$$, a particular realization of the random set $$S_f$$ for the same function $$f$$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)

Then we also have: $$\mbox{E}[T(n)] \sim C\sum_{k=1}^{n-1} f(k)f(k+1)\sim C'\int_2^n \frac{dt}{(\log t)^2} .$$

Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $$S_f$$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?

Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $$n$$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to $$\sum_{k=1}^{\lfloor n/2 \rfloor} f(k)f(n-k) \sim \frac{n}{2(\log n)^2}.$$ See also here and here.

• I wrote an article about twins prime conjecture (k-tuple conjecture in general) without using probability theory : mathoverflow.net/questions/338247/… Sep 29, 2019 at 13:09