Skewes's number

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

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

${\displaystyle e^{e^{e^{79}}}<10^{10^{10^{34}}}.}$

In Skewes (1955), without assuming the Riemann hypothesis, Skewes proved that there must exist a value of ${\displaystyle x}$ below

${\displaystyle e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}.}$

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 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 Lehman (1966), who showed that somewhere between ${\displaystyle 1.53\times 10^{1165}}$ and ${\displaystyle 1.65\times 10^{1165}}$ there are more than ${\displaystyle 10^{500}}$ consecutive integers ${\displaystyle x}$ with ${\displaystyle \pi (x)>\operatorname {li} (x)}$. Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of ${\displaystyle 7\times 10^{370}}$. A better estimate was ${\displaystyle 1.39822\times 10^{316}}$ discovered by Bays & Hudson (2000), who showed there are at least ${\displaystyle 10^{153}}$ consecutive integers somewhere near this value where ${\displaystyle \pi (x)>\operatorname {li} (x)}$. Bays and Hudson found a few much smaller values of ${\displaystyle x}$ where ${\displaystyle \pi (x)}$ gets close to ${\displaystyle \operatorname {li} (x)}$; 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 ${\displaystyle x}$ violating ${\displaystyle \pi (x)<\operatorname {li} (x),}$ below ${\displaystyle e^{727.9513468}<1.39718\times 10^{316}}$. This can be reduced to ${\displaystyle e^{727.9513386}<1.39717\times 10^{316}}$ assuming the Riemann hypothesis. Stoll & Demichel (2011) gave ${\displaystyle 1.39716\times 10^{316}}$.

Rigorously, Rosser & Schoenfeld (1962) proved that there are no crossover points below ${\displaystyle x=10^{8}}$, improved by Brent (1975) to ${\displaystyle 8\times 10^{10}}$, by Kotnik (2008) to ${\displaystyle 10^{14}}$, by Platt & Trudgian (2014) to ${\displaystyle 1.39\times 10^{17}}$, and by Büthe (2015) to ${\displaystyle 10^{19}}$.

There is no explicit value ${\displaystyle x}$ known for certain to have the property ${\displaystyle \pi (x)>\operatorname {li} (x),}$ 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 ${\displaystyle \pi (x)>\operatorname {li} (x)}$ 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 gave an explicit formula for ${\displaystyle \pi (x)}$, whose leading terms are (ignoring some subtle convergence questions)

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

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

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

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

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

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

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

(if such a prime exists) is the Skewes number for ${\displaystyle P.}$

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

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

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

• 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.