What is the Method for Finding the Derivative of a Zeta Function?

[heartless]

Hello,
I was trying recently to find a derivative of a zeta function but finally I failed. Can anyone show me a way to find it, I'm more interested in the way of finding it rather than clear aprox. solution. Thanks,

 

[kryptyk]

If Re > 1 then,

$$\zeta(s) = \sum_{k=1}^{\infty} k^{-s}$$

Can't we just differentiate term by term?

$$\frac{\partial}{\partial s} k^{-s} = -k^{-s} \log k$$

Then we have:

$$\frac{\partial}{\partial s} \zeta(s) = -\sum_{k=1}^{\infty} k^{-s} \log k$$

If Re <= 1 then we could use analytic continuation or we could use an integral definition for the zeta function.

 

[shmoe]

Differentiating term by term is valid, the Dirichlet series is absolutely convergent in any right half plane Re(s)>=1+e, where e>0. You can differentiate the usual functional equation to get involving the derivative, though it has more terms than the usual one.

The same method of analytic continuation for zeta via Euler-Maclaurin summation will work here as well for example, if you're after an expression you'd like to numerically work with.