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Fractional Integration and the Riemann Zeta function

  1. Oct 30, 2005 #1

    benorin

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    Homework Helper

    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? Any help, suggestions, or even wanton derision, well, not that :yuck:, but anything useful would be counted kind.
    Thanks,
    -Ben
     
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