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Functional equation Riemann Zeta function

  1. Oct 15, 2015 #1
    There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here:


    Can someone help me linking Corollary 2 to Theorem 1?

    I am just a amateurish layman who tries to piece things together from various sources from the net. So I got confused when I found out that people use two different forms of the functional equation.

    Any help, hints or links are very much appreciated!
  2. jcsd
  3. Oct 15, 2015 #2


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    Another way to write the functional equation for ##\zeta## in Theorem 1 is

    ## \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s) ##

    so if you take as definition of ## \xi(s) ## the formula (6) in the corollary and using the functional formula above you can see that

    ## \xi(s)=\xi(1-s) ##
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