Proof of the infinitude of primes by probabilistic methods. I'm looking to see if there is proof of the infinitude of prime numbers using a probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between independence of measurable sets and coprime integers.
More precisely I am asking the following.

QUESTION: Is there a proof of the infinitude of primes using the Lovász local lemma by any of several different versions of the lemma?

 A: I don't know how to do it using your proposed lemmas, but if you would like a probabilistic proof, we can work it out from my previous answer. Again take $P(X=n) = n^{-s}/\zeta(s)$
and $E_k := \{X \text{ is divisible by } k\}$.
We already showed for $s>1$
$$
\left(\sum_{n=1}^\infty n^{-s}\right)^{-1}=\frac{1}{\zeta(s)} = P(X=1) = P(\cap_{p} E_p^c) = \prod_p(1-P(E_p)) = \prod_p(1-p^{-s}).
$$
Assume for contradiction that there are finitely many primes.
Now let $s\to 1^+$. Then we get
$$
0 = \prod_p(1-p^{-1})
$$
which cannot be, as the right side is a finite product of strictly positive terms. Thus it must be that there are infinitely many primes.
A: If your question is whether there exists yet a proof that applies the Lovász/Erdős Local Lemma to prove the prime infinitude. The answer is, no. The Lemma provides a method that helps on existence proofs and gives not a motivation towards infinitude of primes.
If I misunderstood your question then I would need to know the exact motivation behind your inspiration.
