# functional equation for the Riemann Xi function

The Riemann Xi Function satisfies the following functional equation:

$$\mathrm{\Xi}(s)=\mathrm{\Xi}(1-s)$$ |

This equation directly implies the Riemann Zeta function^{}’s functional equation.

This equation plays an important role in the theory of the Riemann Zeta function. It allows one to analytically continue the Zeta and the Xi functions^{} to the whole complex plane^{}. The definition of the Zeta function^{} as a series is only valid when $\mathrm{\Re}(s)>1$. By using this equation, one can express the values of these two functions when $$ in terms of the values when $\mathrm{\Re}(s)>1$. As an illustration of its importance, one can cite the fact that there are no zeros of the Zeta function with real part^{} greater than 1, so without this functional equation the study of the Zeta function would be very limited.

Title | functional equation for the Riemann Xi function |
---|---|

Canonical name | FunctionalEquationForTheRiemannXiFunction |

Date of creation | 2013-03-22 13:24:15 |

Last modified on | 2013-03-22 13:24:15 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 11M06 |