It is named for the South African mathematician Stanley Skewes who first computed an upper bound on its value.
The name is sometimes also applied to either of the large number bounds which Skewes found.
Littlewood's proof did not, however, exhibit a concrete such number
Skewes's task was to make Littlewood's existence proof effective: exhibit some concrete upper bound for the first sign change.
According to Georg Kreisel, this was not considered obvious even in principle at the time.
[citation needed] Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number
below Without assuming the Riemann hypothesis, Skewes (1955) later proved that there exists a value of
below These upper bounds have since been reduced considerably by using large-scale computer calculations of zeros of the Riemann zeta function.
The first estimate for the actual value of a crossover point was given by Lehman (1966), who showed that somewhere between
Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of
Bays and Hudson found a few much smaller values of
; the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though computer calculations suggest they are unlikely to exist.
Chao & Plymen (2010) gave a small improvement and correction to the result of Bays and Hudson.
Saouter & Demichel (2010) found a smaller interval for a crossing, which was slightly improved by Zegowitz (2010).
Rigorously, Rosser & Schoenfeld (1962) proved that there are no crossover points below
though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
Rubinstein & Sarnak (1994) showed that this proportion is about 2.6×10−7, which is surprisingly large given how far one has to go to find the first example.
, 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 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.
random complex numbers having roughly the same argument is about 1 in
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 1/2) are eventually larger than
is roughly analogous to a second-order correction accounting for squares of primes.
An equivalent definition of Skewes's number exists for prime k-tuples (Tóth (2019)).