Homework Help: Zeta Function

1. Jan 31, 2010

seanhbailey

1. The problem statement, all variables and given/known data

Does the sum $$\sum_{n=0}^{\infty} zeta^{(n)}(s)$$ 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. Jan 31, 2010

Count Iblis

Re: Zeta Function I NEED AN ASWER PLEASE!

Divergent, but you can still resum the series using Borel resummation.

3. Jan 31, 2010

seanhbailey

Re: Zeta Function I NEED AN ASWER PLEASE!

How do i do that?

4. Jan 31, 2010

seanhbailey

Re: Zeta Function I NEED AN ASWER PLEASE!

Can anyone prove that the sum $$\sum_{n=0}^{\infty} zeta^{(n)}(s)$$ goes to INFINITY not just diverges

5. Feb 1, 2010

Count Iblis

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.