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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 .
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 estimates
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
|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|
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 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.
Equivalent for prime k-tuples
An equivalent definition of Skewes' number exists for prime k-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 (see first 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 the Skewes number for .
The table below shows the currently known Skewes numbers for prime k-tuples:
|Prime k-tuple||Skewes number||Found by|
The Skewes number (if it exists) for sexy primes is still unknown.
- Bays, C.; Hudson, R. H. (2000), "A new bound for the smallest with " (PDF), Mathematics of Computation, 69 (231): 1285–1296, doi:10.1090/S0025-5718-99-01104-7, MR 1752093, Zbl 1042.11001
- Brent, R. P. (1975), "Irregularities in the distribution of primes and twin primes", Mathematics of Computation, 29 (129): 43–56, doi:10.2307/2005460, JSTOR 2005460, MR 0369287, Zbl 0295.10002
- Büthe, Jan (2015), An analytic method for bounding , arXiv:1511.02032, Bibcode:2015arXiv151102032B
- Chao, Kuok Fai; Plymen, Roger (2010), "A new bound for the smallest with ", International Journal of Number Theory, 6 (03): 681–690, arXiv:math/0509312, doi:10.1142/S1793042110003125, MR 2652902, Zbl 1215.11084
- Kotnik, T. (2008), "The prime-counting function and its analytic approximations", Advances in Computational Mathematics, 29 (1): 55–70, doi:10.1007/s10444-007-9039-2, MR 2420864, Zbl 1149.11004
- Lehman, R. Sherman (1966), "On the difference ", Acta Arithmetica, 11: 397–410, doi:10.4064/aa-11-4-397-410, MR 0202686, Zbl 0151.04101
- Littlewood, J. E. (1914), "Sur la distribution des nombres premiers", Comptes Rendus, 158: 1869–1872, JFM 45.0305.01
- Platt, D. J.; Trudgian, T. S. (2014), On the first sign change of , arXiv:1407.1914, Bibcode:2014arXiv1407.1914P
- te Riele, H. J. J. (1987), "On the sign of the difference ", Mathematics of Computation, 48 (177): 323–328, doi:10.1090/s0025-5718-1987-0866118-6, JSTOR 2007893, MR 0866118
- Rosser, J. B.; Schoenfeld, L. (1962), "Approximate formulas for some functions of prime numbers", Illinois Journal of Mathematics, 6: 64–94, MR 0137689
- Saouter, Yannick; Demichel, Patrick (2010), "A sharp region where is positive", Mathematics of Computation, 79 (272): 2395–2405, doi:10.1090/S0025-5718-10-02351-3, MR 2684372
- Rubinstein, M.; Sarnak, P. (1994), "Chebyshev's bias", Experimental Mathematics, 3 (3): 173–197, doi:10.1080/10586458.1994.10504289, MR 1329368
- Skewes, S. (1933), "On the difference ", Journal of the London Mathematical Society, 8: 277–283, doi:10.1112/jlms/s1-8.4.277, JFM 59.0370.02, Zbl 0007.34003
- Skewes, S. (1955), "On the difference (II)", Proceedings of the London Mathematical Society, 5: 48–70, doi:10.1112/plms/s3-5.1.48, MR 0067145
- Stoll, Douglas; Demichel, Patrick (2011), "The impact of complex zeros on for ", Mathematics of Computation, 80 (276): 2381–2394, doi:10.1090/S0025-5718-2011-02477-4, MR 2813366
- Tóth, László (2019), "On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood" (PDF), Computational Methods in Science and Technology, 25 (3).
- Wintner, A. (1941), "On the distribution function of the remainder term of the prime number theorem", American Journal of Mathematics, 63 (2): 233–248, doi:10.2307/2371519, JSTOR 2371519, MR 0004255
- Wolf, Marek (2011), "The Skewes number for twin primes: counting sign changes of π2(x) − C2Li2(x)" (PDF), Computational Methods in Science and Technology, 17.
- Zegowitz, Stefanie (2010), On the positive region of , Master's thesis, Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester
- Demichels, Patrick. "The prime counting function and related subjects" (PDF). Demichel. Archived from the original (pdf) on Sep 8, 2006. Retrieved 2009-09-29.
- Asimov, I. (1976). "Skewered!". Of Matters Great and Small. New York: Ace Books. ISBN 978-0441610723.