The exact value of Skewes's number is still not known, but it is known that there is a crossing between and near It is not known whether this is the smallest crossing.
The name is sometimes also applied to either of thelarge number bounds which Skewes found.
Although nobody has ever found a value of for which Skewes's research supervisorJ.E. Littlewood had proved inLittlewood (1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference changes infinitely many times. Littlewood's proof did not, however, exhibit a concrete such number, nor did it even give any bounds on the value.
Skewes's task was to make Littlewood's existence proofeffective: exhibit some concrete upper bound for the first sign change. According toGeorg Kreisel, this was not considered obvious even in principle at the time.[1]
These upper bounds have since been reduced considerably by using large-scale computer calculations ofzeros of theRiemann zeta function. The first estimate for the actual value of a crossover point was given byLehman (1966), who showed that somewhere between and there are more than consecutiveintegers with.Without assuming the Riemann hypothesis,H. J. J. te Riele (1987) proved an upper bound of. A better estimate was discovered byBays & Hudson (2000), who showed there are at least consecutive integers somewhere near this value where. Bays and Hudson found a few much smaller values of where gets close to; 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 byZegowitz (2010). The same source shows that there exists a number violating below. This can be reduced to assuming the Riemann hypothesis.Stoll & Demichel (2011) conducted an analysis with up to 2×1011 complex zeros which gives computational evidence that a crossover may exist near.
There is no explicit value known for certain to have the property though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
Even though thenatural density of the positive integers for which does not exist,Wintner (1941) showed that thelogarithmic density of these positive integers does exist and is positive.Rubinstein & Sarnak (1994) showed that this proportion is about2.6×10−7, which is surprisingly large given how far one has to go to find the first example.
The largest error term in the approximation (if theRiemann 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 complexarguments, 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 alot smaller than the leading error term, mainly because the firstcomplex zero of the zeta function has quite a largeimaginary 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 ofDirichlet'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 (withreal part greater than1/2) are eventually larger than.
The reason for the term is that, roughly speaking, actually counts powers ofprimes, rather than the primes themselves, with weighted by. The term is roughly analogous to a second-order correction accounting forsquares of primes.
An equivalent definition of Skewes's number exists forprimek-tuples (Tóth (2019)). Let denote a prime (k + 1)-tuple, the number of primes below such that are all prime, let and let denote its Hardy–Littlewood constant (seeFirst Hardy–Littlewood conjecture). Then the first prime that violates the Hardy–Littlewood inequality for the (k + 1)-tuple, i.e., the first prime such that
(if such a prime exists) is theSkewes number for
The table below shows the currently known Skewes numbers for primek-tuples:
^Kreisel (1951) quotes A. E. Ingham (1932) and J. E. Littlewood (1948) as stating "that the proof was believed to be 'non-constructive', or to require 'new ideas' of proof to make it constructive."
Zegowitz, Stefanie (2010),On the positive region of (masters), Master's thesis, Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester
