In 1896, de la Valee Poussin and Hadamard simultaneously proved what had been
suspected for several decades, and what is now known as the
prime number theorem: In words, the (discontinuous) prime counting function π(x) is asymptotic to the (smooth) logarithmic function
x/log x. This means that
the ratio of π(x) to x/log x can be made arbitrarily
close to 1 by considering large enough x. Hence x/log x
provides an approximation of the number of primes less than or equal to x,
and if we take sufficiently large x this approximation can be made as
accurate as we would like (proportionally speaking – very simply, as close to
100% accuracy as is desired). |