# Mertens’ first theorem

For any real number $x\ge 2$ we have

$$\sum _{p\le x}\frac{\mathrm{ln}p}{p}=\mathrm{ln}x+O(1)$$ |

for all prime integers $p$.

Moreover, the term $O(1)$ arising in this formula lies in the open interval $(-1-\mathrm{ln}4,\mathrm{ln}4)$.

Title | Mertens’ first theorem |
---|---|

Canonical name | MertensFirstTheorem |

Date of creation | 2013-03-22 11:46:06 |

Last modified on | 2013-03-22 11:46:06 |

Owner | KimJ (5) |

Last modified by | KimJ (5) |

Numerical id | 8 |

Author | KimJ (5) |

Entry type | Theorem |

Classification | msc 11A25 |

Classification | msc 17B45 |

Classification | msc 17B66 |