Skewes' number

In number theory, Skewes' number is by definition the smallest natural number x for which

π(x) − Li(x) ≥ 0

where π(x) is the prime counting function and Li(x) is the offset logarithmic integral.

J. E. Littlewood proved in 1914 that there is such a number (and so, a first such number); and indeed found that the sign of the difference π(x) − Li(x) changes infinitely often. That such a number exists was not at all clear; in fact, all numerical evidence then available seemed to suggest that π(x) is always less than Li(x). Littlewood's proof did not, however exhibit a concrete such number x; it was not an effective result.

Skewes proved in 1933 that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < Li(x) below

<math>e^{e^{e^{79}}}</math>

(now sometimes called first Skewes' number), which is approximately equal to

<math>10^{10^{8.85 \times 10^{33}}}</math>.

In 1955, without assuming the Riemann hypothesis he managed to prove that there must exist a value of x below

<math>10^{10^{10^{1000}}}</math>

(sometimes called second Skewes' number).

These (enormous) upper bounds have since been reduced considerably. Without assuming the Riemann hypothesis, H. J. J. te Riele in 1987 proved an upper bound of

7 × 10³⁷⁰.

A better estimation was 1.39822 × 10³¹⁶ discovered by Bays and Hudson (2000). The best value for the first crossover is now 1.397162914 × 10³¹⁶ (Demichel 2005). This is with very high confidence the first occurrence of Li(x) < π(x) [1].

Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to George Kreisel, this was at the time not considered obvious even in principle. The approach called unwinding in proof theory looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Skewes's result was celebrated partly because the proof structure used excluded middle, which is not a priori a constructive argument (it divides into two cases, and it isn't computable in which case one is working).

Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.

References

• J.E. Littlewood: "Sur la distribution des nombres premiers", Comptes Rendus 158 (1914), pages 1869-1872
• S. Skewes: "On the difference π(x) − Li(x)", Journal of the London Mathematical Society 8 (1934), pages 277-283
• S. Skewes: "On the difference π(x) − Li(x) (II)", Proceedings of the London Mathematical Society 5 (1955), pages 48-70
• H.J.J. te Riele: "On the difference π(x) − Li(x)", Math. Comp. 48 (1987), pages 323-328

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