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I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome.

Known that Riemann $\zeta$ function is defined as sum over positive integers $n \in \Bbb N$:

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$

Having instead a function $\mathcal N(x)$ such that: $$ \mathcal N(x) = \left\{ \begin{array}{l l} x & \quad \text{if $x\in \Bbb N$}\\ 0 & \quad \text{otherwise} \end{array} \right.$$

how can I formally correct introduce $\mathcal N(x)$ instead of $n$ into the $\zeta$ function formula:

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{?^s}$$

It might be just a trivial question but I can not get it? Do I need probably instead of sum an integral?

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    $\begingroup$ Just write $\displaystyle\sum_{n=1}^\infty\frac{1}{{\cal N}(n)^s}$, because $n={\cal N}(n)$ for all $n\in\Bbb N$. This is a very weird question... $\endgroup$
    – anon
    Jun 14, 2013 at 23:16
  • $\begingroup$ @anon appreciate if you put this into an answer so I would quit. $\endgroup$ Jul 10, 2013 at 7:24

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You can write it like $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{\mathcal{N}(n)^s}. $$ Interestingly, you might be interested to know that this is reminiscent of the Dedekind zeta function, which is defined as a sum over norms of ideals of a number field.

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