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Lets define the following function series (Riemann Zeta Function):

$$\varsigma (z)=\sum _{n=1}^\infty n^{-z} $$

where the power is defined by the primary branch of the Logarithm.

A) prove that the series converges in : $ D= \left\{ z:\operatorname{Re}(z) > 1 \right\}$ and also converges absolutely and uniformly in $D_\delta = \left\{z:\operatorname{Re}(z)>1+ \delta \right\} $

B) find $\varsigma '(z) $.

Do guys have any hints? I thought about calculating the radius of convergence, but how do I do it in this case? and how do I prvoe that it converges with this minus power? also I didnt get the information about the power being defined that way, im not sure also that I translated this accurately from hebrew.

Michael Hardy
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Firas Abd El Gani
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    "Radius of convergence" is a concept that is applicable to power series, in which each term is of the form $c_n z^n.$ But this is a Dirichlet series, in which each term is of the form $c_n n^{-z}.$ The concept of "abscissa of convergence" is applicable here. – Michael Hardy Jun 25 '17 at 19:03
  • Can you show the sequence of analytic functions $f_N(z) = \sum_{n=1}^N n^{-z}=\sum_{n=1}^N e^{-z \log n}$ converges uniformly on $\Re(z) \ge 2$ ? What is $f_N'(z)$ and does it converge (uniformly) ? What about $\Re(z) \ge 1+\delta$ ? – reuns Jun 25 '17 at 19:53
  • @MichaelHardy IMO the linked answer is the worst possible – reuns Jun 25 '17 at 19:54
  • @user1952009 : Does that have some relevance to my comment? Why would I be the addressee of your comment about that? – Michael Hardy Jun 25 '17 at 19:58

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