The problem
Theorem: π(x) ~ li(x) as x → ∞, where π(x) counts primes ≤ x and li(x) = ∫₂ˣ dt/log(t). Equivalently: the n-th prime \(p_{n}\) satisfies \(p_{n}\) ~ n log n.
The Prime Number Theorem
historicReferences
History & significance
Gauss conjectured the logarithmic law at age fifteen (1792-93) from hand calculations; Legendre independently proposed an approximation formula in 1798. Chebyshev (1850-52) proved bounds within a constant factor and introduced the Chebyshev functions ψ and ϑ. Riemann's 1859 memoir reformulated the problem via ζ(s) as a complex-analytic question. The final step required showing ζ(s) ≠ 0 on Re(s) = 1.
The resolution (human proof)
Proven independently in 1896 by Jacques Hadamard and Charles-Jean de la Vallée Poussin, both building on Riemann's memoir. The key step: proving ζ(s) has no zeros on the line Re(s) = 1, achieved via a trigonometric identity argument. De la Vallée Poussin additionally established the zero-free region Re(s) ≥ 1 − c/log(|t|), yielding the error term O(x·exp(−c√log x)). The theorem founded analytic number theory; its quantitative refinements (de la Vallée Poussin, Vinogradov–Korobov) remain active, while RH would sharpen the error to O(√x log x).