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This question already has an answer here:

Here I am considering the original zeta function (not the extended one)$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ When Re(s)>1, the Zeta function converges, and if Re(s)<1 it diverges.

Here is my question. What happens when Re(s)=1?

I know it diverges when s=1, but does it converge otherwise?

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marked as duplicate by Dietrich Burde, Community Jun 1 '16 at 14:32

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    $\begingroup$ No, the series diverges whenever $\operatorname{Re} s = 1$. That's not totally easy to prove, though. $\endgroup$ – Daniel Fischer Jun 1 '16 at 13:38
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    $\begingroup$ Relevant MO thread $\endgroup$ – Daniel Fischer Jun 1 '16 at 13:40