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Fractional Calculus and the Riemann Zeta function

  1. Oct 30, 2005 #1

    benorin

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    So it is well-known that for [tex]n=2,3,...[/tex] the following equation holds

    [tex]\zeta(n)=\int_{x_{n}=0}^{1}\int_{x_{n-1}=0}^{1}\cdot\cdot\cdot\int_{x_{1}=0}^{1}\left(\frac{1}{1-\prod_{k=1}^{n}x_{k}}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}dx_{n}[/tex]

    My question is how can this relation be extended to [tex]n\in\mathbb{C}\setminus \{1\}[/tex], or some appreciable subset thereof (e.g. [tex]\Re(n)>1[/tex] using fractional integration?

    My bad: meant to post this in the Calculus & Analysis forum.
     
    Last edited: Oct 30, 2005
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  3. Oct 31, 2005 #2

    CarlB

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    Maybe you could do it by the "dimensional regularization" method that the physicists use to eliminate cancelling infinities in QFT. The idea is that you write the RHS using a method that works for arbitrary dimensions.

    For the problem at hand, you might begin by converting the limits of integration to go over the whole real line instead of from 0 to 1. Then convert to spherical coordinates and hopefully write it in a way that eliminates N from the number of integrals, for example, by integrating the angular part.

    Do tell us how when you find out.

    Carl
     
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