# Mangoldt summatory function is $O(x)$

###### Theorem 1

$\psi (x)=O(x)$, in other , $\frac{\psi \mathit{}\mathrm{(}x\mathrm{)}}{x}$ is bounded.

Proof.

$$ |

since $$ if $p>\sqrt{x}$. Continuing, we have

$$ |

Note that $\pi (x)\mathrm{ln}x\le 8x\mathrm{ln}2$ by Chebyshev’s bounds on $\pi (x)$ (http://planetmath.org/BoundsOnPin).

Title | Mangoldt summatory function is $O(x)$ |
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Canonical name | MangoldtSummatoryFunctionIsOx |

Date of creation | 2013-03-22 17:42:59 |

Last modified on | 2013-03-22 17:42:59 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 5 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 11A41 |