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Help with capital sigma notation please.

  1. Jun 25, 2012 #1
    I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems.

    The function is defined as follows

    [tex] \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} [/tex]

    where [tex] s \in \mathbb{C} [/tex]

    we know that

    [tex] n^s = exp(s\;ln\;n)[/tex]

    so I can write

    [tex] \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{exp(s\;ln\;n)} [/tex]

    but since

    [tex] \frac{1}{exp(s\;ln\;n)} = 1 + \frac{1!}{(s\;ln\;n)} + \frac{2!}{(s\;ln\;n)^2} + ... =
    1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} [/tex]

    how can I write this ''sum within a sum''? ζ(s) here, if I am correct, would be an infinite sum of terms which are infinite sums.

    Thank you for taking the time to help!


    edit :

    Could I say

    [tex]1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} = a_k[/tex]

    then

    [tex] \zeta (s) = \sum_{k=1}^{\infty} a_k[/tex]

    Does that make any sense?
     
  2. jcsd
  3. Jun 25, 2012 #2

    D H

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    Staff Emeritus
    Science Advisor

    No.

    What you can say is that
    [tex]\exp (s \ln n) = \sum_{r=0}^{\infty} \frac{(s\ln n)^r}{r!}[/tex]
    What you did, simply inverting each term on the right hand side to get [itex]1/\exp(s\ln n)[/itex], is invalid.

    This is what you need to use for [itex]1/\exp(s\ln n)[/itex]:
    [tex]\frac 1{\exp(s\ln n)} = \exp (-s \ln n) =
    \sum_{r=0}^{\infty} \frac{(-s\ln n)^r}{r!} =
    \sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!}[/tex]
     
  4. Jun 25, 2012 #3

    This only makes sense for [itex]\,Re(s)>1\,[/itex]...careful!

    DonAntonio



     
  5. Jun 25, 2012 #4
    Thank you DH for the help, and DonAntonio for your rigor, I need to work on that.

    But in this part,

    [tex]\sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!}[/tex]

    Don't we need to to take the sum of all terms with both n and r, from 1 to ∞? Could I write it as

    [tex]\sum_{n=1}^{\infty} \; \sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!} [/tex] ??
     
    Last edited: Jun 25, 2012
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