Hadamard Product for Riemann's Xi Function

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Discussion Overview

The discussion centers around the infinite product representation of Riemann's Xi function, specifically exploring its relationship with the Gamma function and the Riemann zeta function. Participants are examining the mathematical formulation and manipulation of these functions.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant asks about the infinite product for the function \Xi(s)=\Gamma\left(\frac{s}{2}\right)\pi^{-s/2}\zeta(s).
  • Another participant suggests manipulating Hadamard's zeta function infinite product to derive the answer.
  • A third participant provides a formula involving the product over complex zeros of the zeta function, expressed as {\prod_n (1-s/s_n)\over s(s-1)}.
  • A later reply expresses gratitude towards the participant who provided the formula.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants are presenting different approaches and formulations without resolving the question of the infinite product.

Contextual Notes

Limitations include the dependence on the definitions of the functions involved and the unresolved nature of the infinite product representation.

Who May Find This Useful

Mathematicians and researchers interested in analytic number theory, particularly those studying the properties of the Riemann zeta function and related functions.

Charles49
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What is the infinite product for the function
[tex]\Xi(s)=\Gamma\left(\frac{s}{2}\right)\pi^{-s/2}\zeta(s)[/tex]
?
 
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You can manipulate the Hadamard's zeta function infinite product to get your answer.
 
[tex]{\prod_n (1-s/s_n)\over s(s-1)}[/tex]

where [itex]s_n[/itex] is a complex zero of [itex]\zeta(s)[/itex].
 
Thank you Avodyne
 

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