# Bombieri-Vinogradov theorem

The *Bombieri-Vinogradov theorem*, sometimes called *Bombieri’s theorem*, states that for a positive real number $A$, if ${x}^{\frac{1}{2}}{\mathrm{log}}^{-A}x\le Q\le {x}^{\frac{1}{2}}$ then

$$\sum _{q\le Q}\underset{y\le x}{\mathrm{max}}\underset{\genfrac{}{}{0pt}{}{1\le a\le q}{(a,q)=1}}{\mathrm{max}}\left|\psi (x;q,a)-\frac{x}{\varphi (q)}\right|=O\left({x}^{\frac{1}{2}}Q{(\mathrm{log}x)}^{5}\right),$$ |

where $\varphi (q)$ is Euler’s totient function and

$$\psi (x;q,a)=\sum _{\genfrac{}{}{0pt}{}{n\le x}{n\equiv amodq}}\mathrm{\Lambda}(n),$$ |

where $\mathrm{\Lambda}(n)$ is the Mangoldt function^{}.

Title | Bombieri-Vinogradov theorem |
---|---|

Canonical name | BombieriVinogradovTheorem |

Date of creation | 2013-03-22 16:25:36 |

Last modified on | 2013-03-22 16:25:36 |

Owner | Mravinci (12996) |

Last modified by | Mravinci (12996) |

Numerical id | 4 |

Author | Mravinci (12996) |

Entry type | Theorem |

Classification | msc 11A25 |

Synonym | Bombieri’s theorem |