Skewes number

The Skewes number (according to Stanley Skewes ) is an upper limit for the problem of overestimating the prime number density with the integral logarithm according to Carl Friedrich Gauß . It is an upper bound that holds for a value . In other words, there is a change in sign below the Skewes number , which was predicted by John Edensor Littlewood . ${\displaystyle \pi (x)}$ ${\displaystyle \mathrm {Li} (x)}$ ${\displaystyle x}$ ${\displaystyle \mathrm {Li} (y)<\pi (y)}$ ${\displaystyle y\leq x}$ ${\displaystyle \pi (y)-\mathrm {Li} (y)}$

Skewes found the value for her . The approximation is in use. The Skewes number used to be an example of a particularly large number relevant in mathematics. ${\displaystyle \mathrm {e} ^{\mathrm {e} ^{\mathrm {e} ^{79}}}}$ ${\displaystyle 10^{10^{10^{34}}}}$

The upper bound has been further reduced according to Skewes. In the meantime, the search for a lower bound for the number at which a sign change takes place for the first time is being pushed ahead.

history

The problem of the overestimated prime number density is based on a formula about the distribution of prime numbers that Carl Friedrich Gauß is said to have drawn up at the age of 14 (but he published it much later). Accordingly , the number of prime numbers up to x can be given by the formula ${\displaystyle \pi (x)}$

${\displaystyle \mathrm {Li} (x)=\int _{2}^{x}{\frac {\mathrm {d} t}{\ln t}}}$

be approximated. If one compares with concrete values ​​of , which are determined by means of tables of prime numbers, then is always , and it was believed for a long time, that this applies to all real numbers . ${\displaystyle \mathrm {Li} (x)}$ ${\displaystyle \pi (x)}$ ${\displaystyle \mathrm {Li} (x)>\pi (x)}$ ${\displaystyle x}$

In 1914, JE Littlewood proved that the difference changes the sign infinitely as the x increases . So there must be number ranges in which the Gaussian formula underestimates the prime number density. ${\displaystyle \mathrm {Li} (x)-\pi (x)}$

In 1933 Stanley Skewes, who studied at Littlewood in Cambridge and obtained his doctorate with this work, gave with the number

${\displaystyle 10^{10^{10\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000}}=10^{10^{10^{34}}}}$

a first concrete estimate for an upper limit below which this underestimation occurs for the first time. First he proved this in 1933, assuming the Riemann hypothesis , in the more detailed work in 1955 he was able to lower the limit to (assuming the Riemann hypothesis) and also specify a (higher) upper limit, assuming the Riemann hypothesis was invalid, , sometimes called the "second skewes number". ${\displaystyle \mathrm {e} ^{\mathrm {e} ^{\mathrm {e} ^{7{,}703}}}}$ ${\displaystyle 10^{10^{10^{1000}}}}$

The Skewes number is beyond imagination. GH Hardy called the Skewes number "the greatest number that has ever served a specific purpose in mathematics". If you played chess with all protons of the known universe (at that time approximate ), Hardy calculated that the number of possible "moves" (exchange of the positions of two protons) would roughly correspond to Skewes' number. ${\displaystyle 10^{80}}$

In 1971, Graham's number pushed her from number one. However, this was long after Hardy's death.

In the meantime, Herman te Riele has been able to show that the upper limit for the first underestimation to occur must be below , after Sherman Lehman was able to prove an upper limit of as early as 1966 . Riele also proved that at least consecutive natural numbers between and violate the inequality . The upper limit was improved again by Carter Bays and Richard Hudson in 2000 (they also showed that at least consecutive integers near this number violate the inequality). ${\displaystyle 6{,}69\cdot 10^{370}}$ ${\displaystyle 1{,}65\cdot 10^{1165}}$ ${\displaystyle 10^{180}}$ ${\displaystyle 6{,}62\cdot 10^{370}}$ ${\displaystyle 6{,}69\cdot 10^{370}}$ ${\displaystyle \mathrm {Li} (x)>\pi (x)}$ ${\displaystyle 1{,}39822\cdot 10^{316}}$ ${\displaystyle 10^{153}}$

Lower limits for the first occurrence of the sign change come from JB Rosser and Lowell Schoenfeld ( ), Richard P. Brent ( ), Kotnik 2008 ( ) and Büthe 2015 ( ). ${\displaystyle 10^{8}}$ ${\displaystyle 8\cdot 10^{10}}$ ${\displaystyle 10^{14}}$ ${\displaystyle 10^{19}}$

Aurel Wintner showed in 1941 that the proportion of natural numbers for which the inequality is violated has a positive measure, and M. Rubinstein and Peter Sarnak showed in 1994 that the proportion is around 0.00000026.

literature

• Ralph Boas : The Skewes Number. in Ross Honsberger : Mathematical Plums. Mathematical Association of America 1979, chapter 10.
• Littlewood: A mathematician's miscellany. Methuen 1953, p. 113 f.
• Isaac Asimov : Skewered! Fantasy and Science Fiction. 1974, p. 131 ff. Popular science.

See also

• Exponential representation
• Potency tower
• List of special numbers

Web links

• Eric W. Weisstein : Skewes Number . In: MathWorld (English).

Individual evidence

1. ↑ Littlewood: Sur la distribution des nombres premiers. Comptes Rendus Acad. Sci., Vol. 158, 1914, pp. 1869-1872. Extensively proven in Hardy, Littlewood: Contributions to the theory of the Riemann Zeta Function and the Theory of the Distribution of Primes. Acta Mathematica, Vol. 41, 1918, pp. 119-196. Proof representations can be found in Prachar: Prime Number Distribution . Springer, 1957, Narkiewicz: Development of Prime Number Theory. Springer, 2000, p. 322 ff., Albert Ingham : The distribution of prime numbers. 1932.
2. ↑ Skewes: On the Difference Li (x) - π (x) I. J. London Math. Society, Vol. 8, 1933, pp. 277-283, Part 2, Proc. London Math. Soc. Vol. 5, 1955, pp. 48-70.
3. ↑ The 1933 publication of Skewes was more of a sketch of evidence.
4. ↑ Littlewood also stated in 1937 (Journal of the London Mathematical Society, vol. 12, p. 217) that he had proved an upper limit without assuming the validity of the Riemann hypothesis, but did not publish any proof. In A mathematicians miscellany. Methuen 1953, p. 113, he states that Skewes found such a proof in 1937, but has not yet published it.
5. ↑ Hardy: Ramanujan. 1940, p. 17.
6. ^ Herman te Riele: On the sign of the difference Li (x) - π (x). Mathematics of Computation, Vol. 48, 1987, pp. 323-328.
7. ↑ RS Lehman: On the difference… Acta Arithmetica , Vol. 11, 1966, p. 397. Like te Riele, who used Lehman's method, he proved this without any preconditions.
8. ↑ C. Bays, RH Hudson: A new bound for the smallest x with . ${\displaystyle \pi (x)>\mathrm {Li} (x)}$Mathematics of Computation, Vol. 69, 2000, pp. 1285-1296.
9. ^ Rosser, Schoenfeld: Approximate formulas for some functions of prime numbers. Illinois J. Math., Vol. 6, 1962, pp. 64-94.
10. ^ Mathematics of Computation. Vol. 29, 1975, p. 43.
11. ↑ Kotnik: Advances in Computational Mathematics. Vol. 29, 2008, p. 55.
12. ^ Jan Büthe: An analytic method for bounding ψ (x). 2015, Arxiv.
13. ^ Wintner: On the distribution function of the remainder term of the prime number theorem. American Journal of Mathematics, Vol. 63, 1941, p. 233.
14. ^ Rubinstein, Sarnak: Chebyshevs bias. Experimental Mathematics, Vol. 3, 1994, pp. 173-197. At projecteuclid.org.