Skewes Number

The Skewes number (or first Skewes number) is the number Sk1 above which must fail (assuming that the Riemann Hypothesis is true), where is the Prime Counting Function and is the Logarithmic Integral.

The Skewes number has since been reduced to by te Riele (1987), although Conway and Guy (1996) claim that the best current limit is . In 1914, Littlewood proved that the inequality must, in fact, fail infinitely often.

The second Skewes number Sk2 is the number above which must fail (assuming that the Riemann Hypothesis is false). It is much larger than the Skewes number Sk1,

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 .'' Acta Arith. 11, 397-410, 1966.

te Riele, H. J. J. On the Sign of the Difference .'' Math. Comput. 48, 323-328, 1987.

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