Can someone please explain the concept of a Ramanujan sum in easier language than Wikipedia and its relation to this question. Then, how to calculate the Ramanujan sum:
$$\sum _{n\geq 1}^{\Re } n^{-1/s}$$
To show it equal to:
$$\zeta(1/s)$$
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what is your definition of the Ramanujan sum ? As Alex.R said in the comment, we need to show that $f(z) =\displaystyle\overset{\mathcal{R}}\sum_{n \ge 1} n^{-z}$ is analytic in $z$ – reuns Oct 10 '16 at 13:39
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$$\zeta(z):=\sum_{n=1}^\infty \frac{1}{n^z}.$$
Plug in $z=1/s$.
Alex R.
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Well then it needs to be defined for $z\leq 1$ by analytic continuation. – Alex R. Oct 10 '16 at 00:33
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Do you know what does this have to do with the question in the link? – Ahmed S. Attaalla Oct 10 '16 at 00:35