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Riemann zeta function generalization

  1. Sep 16, 2006 #1
    "Riemann zeta function"...generalization..

    Hello my question is if we define the "generalized" Riemann zeta function:

    [tex] \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s} [/tex]

    which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition to include negative values of "s" (using a functional equation or something similar)..:tongue2: :tongue2:
  2. jcsd
  3. Sep 16, 2006 #2


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    [tex] \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s}=h^{-s} \sum_{n=0}^{\infty}(x/h+n)^{-s}[/tex]

    It's just a Hurwitz zeta function.
  4. Sep 16, 2006 #3

    matt grime

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    There is a whole well documented world of things like this out there, Jose. L functions, generalized zeta functions, indeed the generalized Riemann hypothesis is known to be true for many many of the generalized zeta functions.
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