Skewes number

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There are two distinct Skewes numbers, $Sk_{1}$ and $Sk_{2}$, depending on whether the Riemann Hypothesis is true or false, respectively. The Skewes number is the upper bound $X$, which depends on the truth of RH, for the smallest number $x$ for which the logarithmic integral $li(x)$ is an underestimate of the prime counting function $\pi (x)$, i.e.

$\exists x\leq X\mid \pi (x)>li(x).$

A052435 Round(li(n) - pi(n)), where li is the logarithmic integral and pi(x) is the number of primes up to x. (For n >= 2.) (Here li is the "American" version starting at 0.)

{0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, ...}

RH is true: Sk1

Such a number was proved to exist by J. E. Littlewood in 1912.

In 1933, Stanley Skewes found this number to be

$Sk_{1}=e^{e^{e^{79}}}\approx 10^{10^{10^{34}}}.$

In 1914, Littlewood also proved that $\pi (x)-li(x)$ changes sign infinitely often!

Later on, much smaller upper bounds have been found. Now, the first crossover, if RH is true, is expected to be around 1.397162914 * 10^316 (P. Demichel).

RH is false: Sk2

In 1955, Stanley Skewes found the second Skewes number to be

$Sk_{2}=10^{10^{10^{10^{3}}}}.$

Later on, much smaller upper bounds have been found. Now, the first crossover, if RH is false, is expected to be around 10^1167 (R. Guy and J. Conway).

External links

• Roger Plymen, Skewes Numbers, 2011.
• Weisstein, Eric W., Skewes Number, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Prime Counting Function, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Logarithmic Integral, from MathWorld—A Wolfram Web Resource.