Skewes's number

Unsolved problem in mathematics:

What is the smallest Skewes's number?

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

\pi (x)>\operatorname {li} (x),

where $π$ is the prime-counting function and $li$ is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between $\pi (x)<\operatorname {li} (x)$ and $\pi (x)>\operatorname {li} (x)$ near $e^{727.95133}<1.397\times 10^{316}.$ It is not known whether it is the smallest crossing.

Skewes's numbers

J.E. 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 $\pi (x)-\operatorname {li} (x)$ changes infinitely many times. All numerical evidence then available seemed to suggest that $\pi (x)$ was always less than $\operatorname {li} (x).$ Littlewood's proof did not, however, exhibit a concrete such number $x$ .

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number $x$ violating $\pi (x)<\operatorname {li} (x),$ below

e^{e^{e^{79}}}<10^{10^{10^{34}}}.

Without assuming the Riemann hypothesis, Skewes (1955) proved that there exists a value of $x$ below

e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}.

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 not considered obvious even in principle at the time.

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 $1.53\times 10^{1165}$ and $1.65\times 10^{1165}$ there are more than $10^{500}$ consecutive integers $x$ with $\pi (x)>\operatorname {li} (x)$ . Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of $7\times 10^{370}$ . A better estimate was $1.39822\times 10^{316}$ discovered by Bays & Hudson (2000), who showed there are at least $10^{153}$ consecutive integers somewhere near this value where $\pi (x)>\operatorname {li} (x)$ . Bays and Hudson found a few much smaller values of $x$ where $\pi (x)$ gets close to $\operatorname {li} (x)$ ; 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 $x$ violating $\pi (x)<\operatorname {li} (x),$ below $e^{727.9513468}<1.39718\times 10^{316}$ . This can be reduced to $e^{727.9513386}<1.39717\times 10^{316}$ assuming the Riemann hypothesis. Stoll & Demichel (2011) gave $1.39716\times 10^{316}$ .

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 $x=10^{8}$ , improved by Brent (1975) to $8\times 10^{10}$ , by Kotnik (2008) to $10^{14}$ , by Platt & Trudgian (2014) to $1.39\times 10^{17}$ , and by Büthe (2015) to $10^{19}$ .

There is no explicit value $x$ known for certain to have the property $\pi (x)>\operatorname {li} (x),$ 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 $\pi (x)>\operatorname {li} (x)$ 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 formula

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

\pi (x)=\operatorname {li} (x)-{\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})-\sum _{\rho }\operatorname {li} (x^{\rho })+{\text{smaller terms}}

where the sum is over all $\rho$ in the set of non-trivial zeros of the Riemann zeta function.

The largest error term in the approximation $\pi (x)\approx \operatorname {li} (x)$ (if the Riemann hypothesis is true) is negative ${\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})$ , showing that $\operatorname {li} (x)$ is usually larger than $\pi (x)$ . 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 ${\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})$ .

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 $N$ random complex numbers having roughly the same argument is about 1 in $2^{N}$ . This explains why $\pi (x)$ is sometimes larger than $\operatorname {li} (x),$ 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 $\operatorname {li} (x^{\rho })$ for zeros violating the Riemann hypothesis (with real part greater than ⁠1/2⁠) are eventually larger than $\operatorname {li} (x^{1/2})$ .

The reason for the term ${\tfrac {1}{2}}\mathrm {li} (x^{1/2})$ is that, roughly speaking, $\mathrm {li} (x)$ actually counts powers of primes, rather than the primes themselves, with $p^{n}$ weighted by ${\frac {1}{n}}$ . The term ${\tfrac {1}{2}}\mathrm {li} (x^{1/2})$ is roughly analogous to a second-order correction accounting for squares of primes.

Equivalent for prime k-tuples

An equivalent definition of Skewes' number exists for prime k-tuples (Tóth (2019)). Let $P=(p,p+i_{1},p+i_{2},...,p+i_{k})$ denote a prime (k + 1)-tuple, $\pi _{P}(x)$ the number of primes $p$ below $x$ such that $p,p+i_{1},p+i_{2},...,p+i_{k}$ are all prime, let $\operatorname {li_{P}} (x)=\int _{2}^{x}{\frac {dt}{(\ln t)^{k+1}}}$ and let $C_{P}$ denote its Hardy–Littlewood constant (see First Hardy–Littlewood conjecture). Then the first prime $p$ that violates the Hardy–Littlewood inequality for the (k + 1)-tuple $P$ , i.e., the first prime $p$ such that

\pi _{P}(p)>C_{P}\operatorname {li} _{P}(p),

(if such a prime exists) is the Skewes number for $P.$

The table below shows the currently known Skewes numbers for prime k-tuples:

Prime k-tuple	Skewes number	Found by
(p, p + 2)	1369391	Wolf (2011)
(p, p + 4)	5206837	Tóth (2019)
(p, p + 2, p + 6)	87613571	Tóth (2019)
(p, p + 4, p + 6)	337867	Tóth (2019)
(p, p + 2, p + 6, p + 8)	1172531	Tóth (2019)
(p, p + 4, p +6 , p + 10)	827929093	Tóth (2019)
(p, p + 2, p + 6, p + 8, p + 12)	21432401	Tóth (2019)
(p, p +4 , p +6 , p + 10, p + 12)	216646267	Tóth (2019)
(p, p + 4, p + 6, p + 10, p + 12, p + 16)	251331775687	Tóth (2019)
(p, p+2, p+6, p+8, p+12, p+18, p+20)	7572964186421	Pfoertner (2020)
(p, p+2, p+8, p+12, p+14, p+18, p+20)	214159878489239	Pfoertner (2020)
(p, p+2, p+6, p+8, p+12, p+18, p+20, p+26)	1203255673037261	Pfoertner / Luhn (2021)
(p, p+2, p+6, p+12, p+14, p+20, p+24, p+26)	523250002674163757	Luhn / Pfoertner (2021)
(p, p+6, p+8, p+14, p+18, p+20, p+24, p+26)	750247439134737983	Pfoertner / Luhn (2021)

The Skewes number (if it exists) for sexy primes $(p,p+6)$ is still unknown.

It is also unknown whether all admissible k-tuples have a corresponding Skewes number.

References

Bays, C.; Hudson, R. H. (2000), "A new bound for the smallest $x$ with $\pi (x)>\operatorname {li} (x)$ " (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 $\psi (x)$ , arXiv:1511.02032, Bibcode:2015arXiv151102032B
Chao, Kuok Fai; Plymen, Roger (2010), "A new bound for the smallest $x$ with $\pi (x)>\operatorname {li} (x)$ ", International Journal of Number Theory, 6 (3): 681–690, arXiv:math/0509312, doi:10.1142/S1793042110003125, MR 2652902, Zbl 1215.11084
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Lehman, R. Sherman (1966), "On the difference $\pi (x)-\operatorname {li} (x)$ ", 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 $\theta (x)-x$ , arXiv:1407.1914, Bibcode:2014arXiv1407.1914P
te Riele, H. J. J. (1987), "On the sign of the difference $\pi (x)-\operatorname {li} (x)$ ", 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, doi:10.1215/ijm/1255631807, MR 0137689
Saouter, Yannick; Demichel, Patrick (2010), "A sharp region where $\pi (x)-\operatorname {li} (x)$ 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 $\pi (x)-\operatorname {li} (x)$ ", 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 $\pi (x)-\operatorname {li} (x)$ (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 $\zeta (s)$ complex zeros on $\pi (x)$ for $x<10^{10^{13}}$ ", 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), doi:10.12921/cmst.2019.0000033, S2CID 203836016.
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: 87–92, doi:10.12921/cmst.2011.17.01.87-92, S2CID 59578795.
Zegowitz, Stefanie (2010), On the positive region of $\pi (x)-\operatorname {li} (x)$ (masters), Master's thesis, Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester

External links

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.