I hope I am not asking something trivial but I can not figure it myself...
Does there exist a function $f(x,s)$ that is analytic with respect to $x>0$, if not analytic then at least smooth such that:
$$\int_{n-1}^n f(x,s) \, dx=n^{-s},\ n\in\mathbb{N}$$ $$\left\{\int_0^1 f(x,s) \, dx=1,\int_1^2 f(x,s) \, dx=2^{-s},\int_2^3 f(x,s) \, dx=3^{-s},\int_3^4 f(x,s) \, dx=4^{-s},\int_4^5 f(x,s) \, dx=5^{-s},\dotso\right\}$$
Then, of course, we have: $$\int_0^{\infty } f(x,s) \, dx=\zeta (s),\ \Re(s)>1$$
There are infinitely many such functions if we do not insist on being analytic or smooth - we can easily define some piecewise-defined function.
If the analytic function $f(x,s)$ exists how to find it, at least numerically?
I am aware of some integral representations of $\zeta(s)$ such as:
$$\zeta (s)=\frac{1}{\Gamma (s)} \int_0^{\infty } \frac{x^{s-1}}{e^x-1} \, dx,\ \Re(s)>1$$ $$\zeta (s)=s \int_0^{\infty } \lfloor x\rfloor x^{-s-1} \, dx,\ \Re(s)>1$$ $$\zeta (s)=-s \int_0^{\infty } \{\frac{1}{x}\} x^{s-1} \, dx,\ 0<\Re(s)<1$$
None of them fulfill my requirements and I am unable to modify them so that they would.
Update after @runway44 comment:
Proof of $\zeta(s)$ integral formula:
$$\zeta (s)=\frac{1}{\Gamma (s)} \int_0^{\infty } \frac{x^{s-1}}{e^x-1} \, dx$$ $$\sum _{n=1}^{\infty } e^{-n x}=\frac{1}{e^x-1}$$ $$\int_0^{\infty } \frac{x^{s-1}}{e^x-1} \, dx=\int_0^{\infty } x^{s-1} \sum _{n=1}^{\infty } e^{-n x} \, dx=\sum _{n=1}^{\infty } \int_0^{\infty } e^{-n x} x^{s-1} \, dx$$ $$u=n x,\ du=n\ dx$$ $$\sum _{n=1}^{\infty } \int_0^{\infty } \frac{u^{s-1}}{n^{s-1}} e^{-u} \, \frac{du}{n}=\left(\sum _{n=1}^{\infty } \frac{1}{n^s}\right) \int_0^{\infty } e^{-u} u^{s-1} \, du=\zeta (s) \Gamma (s)$$
For $x >0,\Re(s) > 1$ (and by analytic continuation for all $s\not \in \Bbb{Z}_{\le 1},x\not \in \Bbb{Z}_{\le 0}$)
$$\zeta(x,s)=\sum_{k\ge 0} (k+x)^{-s},\qquad \partial_x \zeta(x,s)=-s\sum_{k\ge 0} (k+x)^{-s-1} $$
$$n^{-s} = \zeta(n,s)-\zeta(n+1,s)=-\int_n^{n+1} \partial_x \zeta(x,s)dx$$
Any other $f$ is of the form $$f(x,s)=-\partial_x \zeta(x,s) + \partial_x (\frac{g(x,s)}{\Gamma(1-s)})$$ with $g$ some function analytic on the relevant domain.