Skewes Number

The Skewes number (or first Skewes number) is the number Sk₁ above which $\pi(n)<{\rm Li}(n)$ must fail (assuming that the Riemann Hypothesis is true), where $\pi(n)$ is the Prime Counting Function and $\mathop{\rm Li}\nolimits (n)$ is the Logarithmic Integral.

$$\hbox{Sk}_1= e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}.$$

The Skewes number has since been reduced to $e^{e^{27/4}}\approx 8.185\times 10^{370}$ by te Riele (1987), although Conway and Guy (1996) claim that the best current limit is $10^{1167}$. In 1914, Littlewood proved that the inequality must, in fact, fail infinitely often.

The second Skewes number Sk₂ is the number above which $\pi(n)<{\rm Li}(n)$ must fail (assuming that the Riemann Hypothesis is false). It is much larger than the Skewes number Sk₁,

$${\rm Sk}_2=10^{10^{10^{10^3}}}.$$

See also Graham's Number, Riemann Hypothesis

References

Asimov, I. ``Skewered!'' Of Matters Great and Small. New York: Ace Books, 1976. Originally published in Magazine of Fantasy and Science Fiction, Nov. 1974.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 63, 1987.

Boas, R. P. ``The Skewes Number.'' In Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 61, 1996.

Lehman, R. S. ``On the Difference $\pi(x)-\mathop{\rm li}(x)$.'' Acta Arith. 11, 397-410, 1966.

te Riele, H. J. J. ``On the Sign of the Difference $\pi(x)-{\rm li}(x)$.'' Math. Comput. 48, 323-328, 1987.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 30, 1991.