# properties of Riemann xi function

The Riemann xi function, defined by

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

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  $\frac{1}{2}<\sigma<\infty$.

(ii). If $t$ is any fixed real number, then $|\xi(\sigma\!+\!it)|$ is decreasing for  $-\infty<\sigma<\frac{1}{2}$.

(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 PropertiesOfRiemannXiFunction 2013-03-22 19:35:25 2013-03-22 19:35:25 pahio (2872) pahio (2872) 7 pahio (2872) Result msc 11M06 RobinsTheorem ExtraordinaryNumber