value known for certain to have the property though computer calculations suggest some explicit numbers that are quite likely to satisfy this. Even though the natural density of the positive integers for which does not exist, showed that the logarithmic density of these positive integers does exist and is positive. showed that this proportion is about 0.00000026, which is surprisingly large given how far one has to go to find the first example. Riemann's formula Riemann gave an explicit formula for , whose leading terms are (ignoring some subtle convergence questions) where the sum is over all in the set of non-trivial zeros of the Riemann zeta function. The largest error term in the approximation (if the Riemann hypothesis is true) is negative , showing that is usually larger than . The other terms above are somewhat smaller, and moreover tend to have different, seemingly random complex arguments, so mostly cancel out. Occasionally however, several of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term . The reason why the Skewes number is so large is that these smaller terms are quite a lot smaller than the leading error term, mainly because the first complex zero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of random complex numbers having roughly the same argument is about 1 in . This explains why is sometimes larger than and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function. The argument above is not a proof, as it assumes the zeros of the Riemann zeta function are random, which is not true. Roughly speaking, Littlewood's proof consists of Dirichlet's approximation theorem to show that sometimes many terms have about the same argument. In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms for zeros violating the Riemann hypothesis (with real part greater than ) are eventually larger than . The reason for the term is that, roughly speaking, actually counts powers of primes, rather than the