I am trying to find values for $z$ in $\zeta(z)$. However I am tired of using the series:

$$\zeta(z)=\sum_{k=1}^\infty \frac{1}{k^z}$$

Is there any other method to find $\zeta(z)$ for $Re(z)>1$?

  • $\begingroup$ Yes, there are many alternative representations of $\zeta$. One of the easiest to obtain directly from the "standard" series representation is the relationship to the $\eta$ function. And the standard series representation of $\eta$ is valid for $\text{Re}(z)>0$! $\endgroup$ – Mark Viola Mar 3 '18 at 18:31
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    $\begingroup$ What do you mean by "find $z$ in $\zeta(z)?$" Do you want to compute the functional inverse $\zeta^{-1}(z)$ or just $\zeta(z)$ for $z>1?$ $\endgroup$ – gammatester Mar 3 '18 at 18:42

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