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In number theory, Skewes's number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number for which

where π is the prime-counting function and li is the logarithmic integral function. There is a crossing near It is not known whether it is the smallest.

Contents

Skewes's numbersEdit

John Edensor Littlewood, who was Skewes's 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   changes infinitely often. All numerical evidence then available seemed to suggest that   was always less than  . Littlewood's proof did not, however, exhibit a concrete such number  .

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

 .

In Skewes (1955), without assuming the Riemann hypothesis, Skewes proved that there must exist a value of   below

 .

Skewes's task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to Georg Kreisel, this was at the time not considered obvious even in principle.

More recent estimatesEdit

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   and   there are more than   consecutive integers   with  . Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of  . A better estimate was   discovered by Bays & Hudson (2000), who showed there are at least   consecutive integers somewhere near this value where   and suggested that there are probably at least  . 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 by Zegowitz (2010). The same source shows that there exists a number   violating   below  . This can be reduced to  , assuming the Riemann hypothesis. Stoll & Demichel (2011) gave  .

Year near x # of complex
zeros used
by
2000 1.39822×10316 1×106 Bays and Hudson
2010 1.39801×10316 1×107 Chao and Plymen
2010 1.397166×10316 2.2×107 Saouter and Demichel
2011 1.397162×10316 2.0×1011 Stoll and Demichel

Rigorously, Rosser & Schoenfeld (1962) proved that there are no crossover points below  , improved by Brent (1975) to  , by Kotnik (2008) to  , by Platt & Trudgian (2014) to  , and by Büthe (2015) 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, Wintner (1941) showed that the logarithmic density of these positive integers does exist and is positive. Rubinstein & Sarnak (1994) showed that this proportion is about 0.00000026, which is surprisingly large given how far one has to go to find the first example.

Riemann's formulaEdit

Riemann gave an explicit formula for π(x), whose leading terms are (ignoring some subtle convergence questions)

 

where the sum is over zeros ρ of the Riemann zeta function. The largest error term in the approximation   (if the Riemann hypothesis is true) is  , showing that   is usually larger than  . The other terms above are somewhat smaller, and moreover tend to have different complex arguments so mostly cancel out. Occasionally however, many 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. 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 (with real part greater than 1/2) are eventually larger than  .

The reason for the term   is that, roughly speaking,   counts not primes, but powers of primes   weighted by  , and   is a sort of correction term coming from squares of primes.

ReferencesEdit

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