# prime counting function

The prime counting function is a non-multiplicative function for any positive real number $x$, denoted as $\pi (x)$ and gives the number of primes not exceeding $x$. It usually takes a positive integer $n$ for an argument. The first few values of $\pi (n)$ for $n=1,2,3,\mathrm{\dots}$ are $0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8\mathrm{\dots}$ (http://www.research.att.com/ njas/sequences/eisA.cgi?Anum=000720OEIS A000720 ).

The asymptotic behavior of $\pi (x)\sim x/\mathrm{ln}x$ is given by the prime number theorem^{}. This function is closely related with Chebyshev’s functions $\vartheta (x)$ and $\psi (x)$.

Title | prime counting function |
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Canonical name | PrimeCountingFunction |

Date of creation | 2013-03-22 12:49:00 |

Last modified on | 2013-03-22 12:49:00 |

Owner | XJamRastafire (349) |

Last modified by | XJamRastafire (349) |

Numerical id | 13 |

Author | XJamRastafire (349) |

Entry type | Definition |

Classification | msc 11A25 |

Classification | msc 11A41 |

Classification | msc 11N05 |

Related topic | LogarithmicIntegral2 |