properties of Riemann xi function
The Riemann xi function, defined by
is an entire function having as zeros the nonreal zeros of the Riemann zeta function
and only them.
The modulus of the xi function is strictly increasing along every horizontal half-line lyingin any open right half-plane that contains no xi zeros. As well, the modulus decreases strictly alongevery 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 is any fixed real number, then is increasing for .
(ii). If is any fixed real number, then 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.