Let π(x) denote the number of primes less than x, and let li(x) denote the logarithmic integral of x, that is, The prime number theorem, proved independently by Hadamard and La Vallée Poussin in 1896, states that π(x) and li(x) are asymptotically equivalent as x tends to infinity. Early calculations seem to suggest that π(x) < li(x) always holds. But in 1914, the English mathematician John Edensor Littlewood proved that the difference π(x) − li(x) changes sign infinitely often.
In 1859, Riemann had observed that the distribution of prime numbers is linked to the behavior of the zeta function, given by ζ(s) = 1 + 1/2s + 1/3 s + 1/4s + ...
The Riemann hypothesis states that all the relevant solutions of the equation
ζ(s) = 0 are complex and lie on the line Re(s) = 1/2. To this day, this hypothesis, which therefore has major consequences for prime numbers, remains unproved. It is one of the seven Millennium Prize Problems, with a prize of one million dollars offered for its solution.
Stanley Skewes, a student of Littlewood's, proved in 1933, assuming the Riemann hypothesis, that there is an x less than (a value slightly less than ) such that π(x) > li(x).
This bound is known as the first Skewes number. Then, in 1955, without using the Riemann hypothesis, he gave another, larger bound, known as the second Skewes number. Since then, these values, which can fairly be described as gigantic, have been reduced dramatically.
