Prime number theorem
Teorema de los números primos
- Date
- 1896 · confidence high
- Place
- Paris and Leuven
- Credited to
- Jacques Hadamard and Charles-Jean de la Vallée Poussin (independent)
- People
- 👤 Charles-Jean de la Vallée Poussin1866–1962👤 Jacques Hadamard1865–1963
- Field
- Knowledge & symbols
⚠️ Disputed: the sources disagree.
What was discovered
The number of primes less than x is asymptotically x / ln x, as conjectured by Gauss and Legendre. Proved by showing that zeta does not vanish on the line of real part 1, it is the greatest success of 19th-century analytic number theory.
A simultaneous and independent discovery; in 1949 Selberg and Erdős gave an elementary proof amid a bitter priority dispute.
What it led to
- analytic number theory
- prime density estimates for cryptography
- the elementary proofs of Selberg and Erdős (1949)
Built on
- Divergence of the sum of the reciprocals of the primesc. 1737
- Riemann hypothesis1859
- Chebyshev (bounds, 1850)
Made possible
- RSA key generation (abundance of primes)