One significant consequence of the prime number theorem, as it was proved by Charles de la Vallée Poussin and Jacques Hadamard in 1896, that π(n) ~ n/ln(n) is that the average density of prime numbers in the range of 1 to n must be 1/ln n, so asymptotically, the probability of some randomly chosen number in the neighborhood of n being prime is approximately 1/ln n. This might well be considered as an asymptotic probability density function for the prime numbers, so one can derive an approximate probability distribution function for the prime numbers (which is essentially another approximation to π(x)) by taking the integral of 1/ln n. That integral cannot be evaluated in terms of elementary functions, so it is a special function known as the logarithmic integral function Li(x). It can also be shown that as n-> ∞ Li(n), -> n/ln n, so another way of stating the prime number theorem would be to say π(n) ~ Li(n). While n/ln n is a lower bound, it may be shown that Li(n) is more or less an upper bound to the prime counting function π(n). Further, it may be shown that it is an asymptotically tighter bound on π(n) than n/ln n.

This statement of the prime number theorem is more common in analytical number theory, and the study of how much the approximation of Li(x) diverges from π(x) leads straight into the Riemann hypothesis and some of its consequences. Some initial results were achieved by John Littlewood in 1914, who showed that Li(x) is not a strict upper bound on π(x), but that at some point it would cross over and the difference Li(x) - π(x) must alternate back and forth between positive and negative. The point at which these Littlewood violations begin to happen is not known, but one of Littlewood's students, Samuel Skewes, showed that if the Riemann hypothesis is true. the first violations must come before e^{e^(e^79)}, a monstrous number which is about 10^{10^(10^34)} that attained notoriety as Skewes number, the largest number to arise out of a mathematical proof at the time (right know I think Graham's number holds that dubious honor). In 1955 Skewes improved his results, this time without assuming the truth of the Riemann hypothesis, and found that Littlewood violations should begin occurring after 10^{10^10000}. Further refinements were made in the following decades, of which the most recent is the work in 2000 of Carter Bays and Richard Hudson that showed that the most likely region for the first few violations must be in the vicinity of 1.39822 x 10^{316}. Nobody has computed all the prime numbers up to that point just yet, and no one has actually proved the Riemann hypothesis yet either, so it's still a wide open question.