Skewes' number
In number theory, Skewes' number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
where π is the prime-counting function and li is the logarithmic integral function. These bounds have since been improved by others: there is a crossing near . It is not known whether it is the smallest.
John Edensor Littlewood, who was Skewes' research supervisor, had 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) was always less than li(x). 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 was able 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.
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