x
\pi(x)>\operatorname{li}(x),
\pi(x)<\operatorname{li}(x)
\pi(x)>\operatorname{li}(x)
e727.95133<1.397 x 10316.
J.E. Littlewood, who was Skewes's research supervisor, had proved in 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)
\pi(x)
\operatorname{li}(x).
x
proved that, assuming that the Riemann hypothesis is true, there exists a number
x
\pi(x)<\operatorname{li}(x),
| |||||
e |
| |||||
<10 |
.
Without assuming the Riemann hypothesis, proved that there exists a value of
x
| |||||||||
e |
| |||||
<10 |
.
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.
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
1.53 x 101165
1.65 x 101165
10500
x
\pi(x)>\operatorname{li}(x)
7 x 10370
1.39822 x 10316
10153
\pi(x)>\operatorname{li}(x)
x
\pi(x)
\operatorname{li}(x)
x
\pi(x)<\operatorname{li}(x),
e727.9513468<1.39718 x 10316
e727.9513386<1.39717 x 10316
1.39716 x 10316
Year | near x |
| by |
---|---|---|---|
2000 | 1.39822 | 1 | Bays and Hudson |
2010 | 1.39801 | 1 | Chao and Plymen |
2010 | 1.397166 | 2.2 | Saouter and Demichel |
2011 | 1.397162 | 2.0 | Stoll and Demichel |
Rigorously, proved that there are no crossover points below
x=108
8 x 1010
1014
1.39 x 1017
1019
There is no explicit value
x
\pi(x)>\operatorname{li}(x),
Even though the natural density of the positive integers for which
\pi(x)>\operatorname{li}(x)
Riemann gave an explicit formula for
\pi(x)
\pi(x)=\operatorname{li}(x)-\tfrac{1}{2}\operatorname{li}(\sqrt{x})-\sum\rho\operatorname{li}(x\rho)+smallerterms
\rho
The largest error term in the approximation
\pi(x) ≈ \operatorname{li}(x)
\tfrac{1}{2}\operatorname{li}(\sqrt{x})
\operatorname{li}(x)
\pi(x)
\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
2N
\pi(x)
\operatorname{li}(x),
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)
\operatorname{li}(x1/2)
The reason for the term
\tfrac{1}{2}li(x1/2)
li(x)
pn
1 | |
n |
\tfrac{1}{2}li(x1/2)
An equivalent definition of Skewes' number exists for prime k-tuples . Let
P=(p,p+i1,p+i2,...,p+ik)
\piP(x)
p
x
p,p+i1,p+i2,...,p+ik
\operatorname{liP}(x)=
x | |
\int | |
2 |
dt | |
(lnt)k+1 |
CP
p
P
p
\piP(p)>CP\operatorname{li}P(p),
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 | ||
(p, p + 4) | 5206837 | ||
(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)
It is also unknown whether all admissible k-tuples have a corresponding Skewes number.