Primes as uncorrelated random variables [closed]

Question

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?

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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 $\forall t, \sum_{i=1}^{n}P(X_{i,t}=1)=1$.

Answer 1

As I understand your question, we have random variables $X_n$, $n\in\mathbb N$ taking values in $\{0,1\}$.

We intuitively think of $X_n=1$ as ``$n$ is prime'', but other than that this has little to do with primes.

Letting $p_n:=\Pr(X_n=1)$, we have that heuristically $p_n\rightarrow 0$ at a certain rate, and we want to conclude something about the number of $X_k$'s that are $=1$ for $k\le n$.

Note that if $X_i$ and $X_j$ are uncorrelated, then they are (pairwise) independent. (This is true for any random variables that take values in $\{0,1\}$).

Moreover, the Law of Large Nunbers only requires pairwise independence.

[It may be that that fact breaks down for non-constant $\{p_n\}_{n\in\mathbb N}$ but I guess it should be okay.]

Thus, it seems the same informal argument does work with uncorrelatedness in place of independence.

Answer 2

Although this is a good heuristic, I have to add it doesn't capture the number-theoretic nature of the problem. One way to see this is that the same heuristic would imply every odd number is the sum of two primes, which unfortunately has more counterexamples than examples. The reason is that the primes are not random at all in terms of congruences (or, using more jargon, in finite places). One extreme example is that almost all primes (except 2, to be precise) are odd, so if you want two primes to add to an odd number, one of them has to be 2, which screws up the probabilistic heuristic quite badly. Thus we need to take congruence relations into account even if we are heuristically guessing the number of representations of a number as a sum of two primes, which explains why our best guess is not $x/log^2x$, but a constant multiple of it (more precisely, the twin prime constant).