Skewes' number
The Skewes number (after Stanley Skewes ) is an upper bound for the problem of estimating the density of primes with the logarithmic integral by Carl Friedrich Gauss. It is an upper bound of X that is valid for a value. In other words, find below the Skewes number rather than a change in sign of which had been predicted by John Edensor Littlewood.
Skewes found the value for them. The approximation is in use. The Skewes number was considered earlier as an example of a particularly large number of relevant in mathematics.
The upper bound has been further reduced by Skewes. Meanwhile, the search for a lower bound on the number at the first time held a sign change is being driven.
History
The problem of over-estimated density of primes is based on a formula concerning the distribution of prime numbers, the Carl Friedrich Gauss is said to have erected at the age of 14 years ( he published but much later ). Accordingly, the number of the prime numbers to x by the formula
Be approximated. Comparing with concrete values of which one determined on the basis of prime tables, as always, and it was long believed that that is true for all real numbers.
In 1914, JE Littlewood proved that the difference changes sign infinitely often becomes larger x. So there must be speed ranges in which the Gaussian formula underestimates the density of primes.
1933 was Stanley Skewes, who studied at Littlewood in Cambridge and his doctorate in this work with him, with the number
A first concrete estimate of an upper limit, below which this underestimation occurs for the first time. First, he proved in 1933 assuming the Riemann hypothesis, in the more detailed work in 1955 he was the limit to decrease (assuming the Riemann hypothesis) and also assuming the non- validity of the Riemann hypothesis specify a (higher ) upper limit, , sometimes also called " second Skewes number."
The Skewes number is beyond imagination. GH Hardy called the Skewes number " the largest number which has ever served a purpose in mathematics ". You played chess with all the particles in the known universe ( approximately ), so figured Hardy before, correspond to the number of possible moves in about Skewes ' number.
In 1971 it was replaced by Graham number of square 1. However, this was long after Hardy's death.
Meanwhile, could te Riele by Herman be shown that the upper limit for the first occurring underestimation below must occur after Sherman Lehman in 1966 could prove an upper limit of. In addition, te Riele proved that violate at least consecutive natural numbers between and inequality. The upper limit has been further improved by Carter Bays and Richard Hudson 2000 ( also they showed that at least consecutive integers close to this number, the inequality hurt ).
Lower limits for the first occurrence of the sign change as submitted by JB Rosser and Lowell Schoenfeld ( ), Richard P. Brent () and Kotnik 2008 ( ).
Aurel Wintner showed in 1941 that the proportion of natural numbers for which the inequality is violated, has positive measure, and M. Rubinstein and Peter Sarnak showed in 1994 that the proportion is approximately 0.00000026.