The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that

*the number of twin primes below $x$ should be roughly $\dfrac{x}{\log^{2} x}$ up to a multiplicative constant*

is just heuristics, because of the lack of independence of the "random variables" considered therein. But what if we considered them as just **uncorrelated** instead of truly **independent**?

Could the argument be still used or not?

Edit: several people have said that the notion of probability here is rather irrelevant, cause the primes form a deterministic system. My question now is thus: in some dynamical system framework, would the set of congruences that are verified in the case of primes force any subset of the positive integers with the same natural density as the one given by the PNT to be the one of a subset of the primes? The idea is to make a random process evolve under constraints until it becomes entirely determined. I apologize for the vagueness of the question, I just try to make my intuitions as 'rigorizable' as possible. To make this more clear, let's consider the double sequence of random variables $X_{i,t}$ with $1\leq i\leq n$ and $t\in\mathbb{N}_{0}$ such that there exists $K_{i,t}>0$ such that $$P(X_{i,t}=1)=K_{i,t}\dfrac{\pi(n)}{n}\left(\prod_{k=1}^{t}(i \mod p_{k})\right)^{1_{t\gt 0}}$$ and $\forall t, \sum_{i=1}^{n}P(X_{i,t}=1)=1$.

idea. :) That it is not (at this time in history) something that generates causality, that is, that gives true proof, is disappointing, but a contingent fact. $\endgroup$ – paul garrett Apr 17 '15 at 0:406more comments