# properties of Riemann xi function

The Riemann xi function, defined by

$\xi (s):={\displaystyle \frac{s}{2}}(s-1){\pi}^{-\frac{s}{2}}\mathrm{\Gamma}\left({\displaystyle \frac{s}{2}}\right)\zeta (s),$ |

is an entire function^{} having as zeros the nonreal zeros of the Riemann zeta function^{} $\zeta $ and only them.

The modulus^{} of the xi function is strictly increasing along every horizontal half-line lying
in any open right half-plane that contains no xi zeros. As well, the modulus decreases strictly along
every horizontal half-line in any zero-free, open left half-plane.

Taking into account the functional equation

$$\xi (1-s)=\xi (s)$$ |

it follows the reformulation of the Riemann hypothesis:

Theorem. The following three statements are equivalent.

(i). If $t$ is any fixed real number, then $|\xi (\sigma +it)|$ is increasing for $$.

(ii). If $t$ is any fixed real number, then $|\xi (\sigma +it)|$ is decreasing for $$.

(iii). The Riemann hypothesis is true.

## References

- 1 Jonathan Sondow & Christian Dumitrescu: A monotonicity property Riemann’s xi function and a reformulation of the Riemann Hypothesis. – Periodica Mathematica Hungarica 60 (2010) 37–40. Also available http://arxiv.org/ftp/arxiv/papers/1005/1005.1104.pdfhere.

Title | properties of Riemann xi function |
---|---|

Canonical name | PropertiesOfRiemannXiFunction |

Date of creation | 2013-03-22 19:35:25 |

Last modified on | 2013-03-22 19:35:25 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 11M06 |

Related topic | RobinsTheorem |

Related topic | ExtraordinaryNumber |