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Homework Help: Zeta Function

  1. Jan 31, 2010 #1
    1. The problem statement, all variables and given/known data

    Does the sum [tex]\sum_{n=0}^{\infty} zeta^{(n)}(s)[/tex] converge regardless of s, where n is the nth derivative of the Riemann Zeta function? If it converges tell what value it converges to.

    2. Relevant equations



    3. The attempt at a solution

    I used the integral test, and I think it diverges. Plus, by plotting the sequence on Matematica, it looks like it is diverging. Am I correct?
     
  2. jcsd
  3. Jan 31, 2010 #2
    Re: Zeta Function I NEED AN ASWER PLEASE!

    Divergent, but you can still resum the series using Borel resummation.
     
  4. Jan 31, 2010 #3
    Re: Zeta Function I NEED AN ASWER PLEASE!

    How do i do that?
     
  5. Jan 31, 2010 #4
    Re: Zeta Function I NEED AN ASWER PLEASE!

    Can anyone prove that the sum [tex]\sum_{n=0}^{\infty} zeta^{(n)}(s)[/tex] goes to INFINITY not just diverges
     
  6. Feb 1, 2010 #5
    Re: Zeta Function I NEED AN ASWER PLEASE!

    Apply the operator 1 - d/ds to the summation. To investigate convergence, you can apply it to the partial sums of the first n terms. If you apply it formally to the infinite summation then you get the same result you would get after performing the Borel resummation.
     
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