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?