The Riemann zeta function is the function of the complex variable $s=α+iβ$, defined in the half-plane $α>1$ by the absolutely convergent series
$$ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$
In many books, the authors speak about the derivative $ζ′(s)$ in the critical strip. My question is: How we can calculate this derivative in despite that $ζ(s)$ is defined in the half-plane $α>1$, thus no convergence in the critical strip.
On
In addition to Arthur's excellent answer, if you look at Edwards's great book , "Riemann's Zeta Function," p. 10-11, you can see $\zeta(s)$ defined as an integral which "remains valid for all $s$"
$$\zeta (s) = \frac{\Pi(- s)}{2 \pi i} \int_{+ \infty}^{+ \infty} \frac{(- x)^s}{e^x - 1}\frac{dx}{x}$$
and for Re($s$) > $1$ is equal to the Dirichlet series you present above.
Here is a link: http://books.google.com/books?id=ruVmGFPwNhQC&pg=PA10
The function is defined everywhere else in the complex plane by analytic continuation, which is a technique to extend in a natural way any function that can be defined as a power series that converges in some open subset of the complex plane.
The technique is based on a theorem of complex analysis that says that any two analytic functions (i.e. they can be expressed as convergent power series) that are equal at infinitely many points in a bounded area of the complex plane are the same everywhere they are defined. Which means that if we have a function $f$ that we know is analytic on for instance some open subset $U\subset\Bbb C$, then there is in some sense at most one function $g$ which is analytic on (almost all of) $\Bbb C$, and at the same time agrees with $f$ when restricted to $U$.
Thus the series $\zeta(s)$ you quote, which is an analytic function on the part of the complex plane where the real part exceeds $1$, can be continued uniquely to a function defined almost everywhere on $\Bbb C$. And that continuation is what we call the Riemann zeta function.