At the start of the 19th century, Legendre and Gauss conjectured that as formula_22 tends to infinity, the number of primes up to formula_22 is asymptotic to formula_24, where formula_25 is the natural logarithm of formula_22. A weaker consequence of this high density of primes was Bertrand's postulate, that for every formula_27 there is a prime between formula_1 and formula_29, proved in 1852 by Pafnuty Chebyshev. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.