The Prime Number Theorem

Let π(x) be the number of primes ≤ x. The Prime Number Theorem (proved 1896 by Hadamard and de la Vallée Poussin) states π(x) ~ x / ln x. This means the density of primes near x is about 1/ln x. For x = 10⁶, ln x ≈ 13.8, so density ~7.2% – actual π(10⁶)=78,498, ratio 7.85%. The proof uses complex analysis and the Riemann zeta function; it was a major triumph of 19th‑century mathematics. Without it, we’d have no quantitative understanding of prime distribution.