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单词 PropertiesOfRiemannXiFunction
释义

properties of Riemann xi function


The Riemann xi function, defined by

ξ(s):=s2(s-1)π-s2Γ(s2)ζ(s),

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 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

ξ(1-s)=ξ(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 |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.

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.
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更新时间:2025/5/4 6:54:40