# Difference between revisions of "Prime number theorem"

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− | + | '''Prime number theorem''' is the name given to several theorems that provide estimates of the number of primes less than or equal to any given number: | |

− | Let π(''n'') be a function providing the number of primes less than or equal to ''n'', for any positive number ''n''. The prime number theorem states that | + | Let π(''n'') be a function providing the number of primes less than or equal to ''n'', for any positive number ''n''. The simplest form of the prime number theorem states that |

− | : <math>\pi(n) | + | :<math>\pi(n)\sim\frac{n}{\ln n}</math>. |

+ | |||

+ | That is, as n tends to infinity, the [[relative error]] between π(''n'') and ''n''/(ln ''n'') tends to zero. This can be expressed using limit notation as | ||

+ | |||

+ | :<math>\lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1,</math> | ||

[[category:mathematics]] | [[category:mathematics]] |

## Revision as of 06:19, 31 August 2007

**Prime number theorem** is the name given to several theorems that provide estimates of the number of primes less than or equal to any given number:

Let π(*n*) be a function providing the number of primes less than or equal to *n*, for any positive number *n*. The simplest form of the prime number theorem states that

- .

That is, as n tends to infinity, the relative error between π(*n*) and *n*/(ln *n*) tends to zero. This can be expressed using limit notation as