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Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series diverges? Thank you very much.

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    $\begingroup$ Answer to this can be googled in a second. $\endgroup$ – Harto Saarinen Oct 21 '14 at 16:45
  • $\begingroup$ Is $|z|\geq 1$? $\endgroup$ – Mark Oct 21 '14 at 16:52
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The Riemann zeta function or Euler–Riemann zeta function, $ζ(s)$, is a function of a complex variable s that analytically continues the sum of the infinite series $$\sum_{n=1}^\infty\frac{1}{n^s}$$ which converges when the real part of s is greater than 1

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    $\begingroup$ You might say why this is true. $\endgroup$ – user98602 Oct 21 '14 at 16:51
  • $\begingroup$ I try a condition on modulus... $\endgroup$ – Mark Oct 21 '14 at 16:53

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