The product converges to $\frac{\pi^2}{6}$, which reveals the distribution of primes.

The product diverges to infinity because the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, proving there are infinitely many primes.

The product equals $0$, showing that primes are infinitely sparse.

The product converges to a finite value, implying that the number of primes is bounded.