Skewes's number

In number theory, Skewes's number is the smallest natural number for which the prime-counting function exceeds the logarithmic integral function It is named for the South African mathematician Stanley Skewes who first computed an upper bound on its value.
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 the large number bounds which Skewes found.

Skewes's bounds

Although nobody has ever found a value of for which Skewes's research supervisor J.E. Littlewood had proved in that there is such a 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 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.
proved that, assuming that the Riemann hypothesis is true, there exists a number violating below
Without assuming the Riemann hypothesis, later proved that there exists a value of below

More recent bounds

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, who showed that somewhere between and there are more than consecutive integers with.
Without assuming the Riemann hypothesis, proved an upper bound of. A better estimate was discovered by, 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. gave a small improvement and correction to the result of Bays and Hudson. found a smaller interval for a crossing, which was slightly improved by. The same source shows that there exists a number violating below. This can be reduced to assuming the Riemann hypothesis. conducted an analysis with up to 2 complex zeros which gives computational evidence that a crossover may exist near.

Year	# of complex zeros used	by	Interval	# of consecutive integers with given
2000		Bays and Hudson		> 1
2010		Chao and Plymen		1
2010	2.2	Saouter and Demichel		6.6587 1.2741
2010	2.2	Zegowitz		6.695531258 1.15527413

Rigorously, proved that there are no crossover points below, improved by to, by to, by to, and by to.
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 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, 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
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 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 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 are eventually larger than.
The reason for the term is that, roughly speaking, actually counts powers of primes, rather than the primes themselves, with weighted by. The term is roughly analogous to a second-order correction accounting for squares of primes.

Equivalent for prime ''k''-tuples

An equivalent definition of Skewes's number exists for prime k-tuples. Let denote a prime -tuple, the number of primes below such that are all prime, let and let denote its Hardy–Littlewood constant. Then the first prime that violates the Hardy–Littlewood inequality for the -tuple, i.e., the first prime such that
is the Skewes number for
The table below shows the currently known Skewes numbers for prime k-tuples:

Prime k-tuple	Skewes number	Found by
	1369391
	5206837
	87613571	Tóth
	337867	Tóth
	1172531	Tóth
	827929093	Tóth
	21432401	Tóth
	216646267	Tóth
	251331775687	Tóth
	7572964186421	Pfoertner
	214159878489239	Pfoertner
	1203255673037261	Pfoertner / Luhn
	523250002674163757	Luhn / Pfoertner
	750247439134737983	Pfoertner / Luhn

The Skewes number for sexy primes is still unknown.
It is also unknown whether all admissible k-tuples have a corresponding Skewes number.