Maths Bite: Skewes’ Number

Skewes’ Number:

latex.png

Skewes’ number is the number above which

\pi(x) > \operatorname{li}(x),

where π(x) is the prime-counting function and li(x) is logarithmic integral function.

Prime Counting Function: is the function counting the number of prime numbers less than or equal to some real number x.

Logarithmic Integral Function: used in prime number theorem as an estimate of the number of prime numbers less than a given value.

File:Logarithmic integral function.svg

In 1912, it was proved by Littlewood that this limit existed, and it – Skewes’ number – was subsequently found by Stanley Skewes, South African mathematician, in 1933.

 

However this limit has been improved to Inline6.gif  by Lehman in 1966.

Numberphile Video:

Hope you enjoyed this quick maths bite! M x

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