properties of Riemann xi function

The Riemann xi function, defined by


is an entire functionMathworldPlanetmath having as zeros the nonreal zeros of the Riemann zeta functionMathworldPlanetmath ζ and only them.

The modulusMathworldPlanetmathPlanetmathPlanetmath 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


it follows the reformulation of the Riemann hypothesis:

Theorem.  The following three statements are equivalent.

(i). If t is any fixed real number, then |ξ(σ+it)| is increasing for  12<σ<.

(ii). If t is any fixed real number, then |ξ(σ+it)| is decreasing for  -<σ<12.

(iii). The Riemann hypothesis is true.


  • 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
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