Skewes' Number - Skewes' Numbers

Skewes' Numbers

John Edensor Littlewood, Skewes' teacher, proved (in (Littlewood 1914)) that there is such a number (and so, a first such number); and indeed found that the sign of the difference π(x) − li(x) changes infinitely often. All numerical evidence then available seemed to suggest that π(x) is always less than li(x), though mathematicians familiar with Riemann's work on the Riemann zeta function would probably have realized that occasional exceptions were likely by the argument given below (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however, exhibit a concrete such number x.

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < li(x) below

In (Skewes 1955), without assuming the Riemann hypothesis, Skewes managed to prove that there must exist a value of x below

Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to George Kreisel, this was at the time not considered obvious even in principle. The approach called unwinding in proof theory looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Although both Skewes' numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.