# Riemann $\mathrm{\Xi}$ function

The *Riemann Xi function*

$$\mathrm{\Xi}(s)={\pi}^{-\frac{1}{2}s}\mathrm{\Gamma}(\frac{1}{2}s)\zeta (s),$$ |

(where $\mathrm{\Gamma}(s)$ is Euler’s Gamma function^{} and $\zeta (s)$ is the Riemann zeta function^{}), is the key to the functional equation for the Riemann zeta function.

Riemann himself used the notation of a lower case xi ($\xi $). The famous Riemann hypothesis is equivalent to the assertion that all the zeros of $\xi $ are real, in fact Riemann himself presented his original hypothesis in terms of that function^{}.

Riemann’s lower case xi is defined as

$$\xi (s)=\frac{1}{2}s(s-1)\mathrm{\Xi}(s).$$ |

Title | Riemann $\mathrm{\Xi}$ function |
---|---|

Canonical name | RiemannXiFunction |

Date of creation | 2013-03-22 13:24:06 |

Last modified on | 2013-03-22 13:24:06 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 11 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11M06 |