Functions $f(x)=\lfloor x\rfloor$ and $g(s)=\frac{\zeta (s)}{s}$ are related by Mellin inversion theorem, for $c>1$, $\Re(s)>1$.
$$\mathcal{M}_x(f(x))(s)=\mathcal{M}_s^{-1}(g(s))(x)$$
$$\tag{1.1}f(x)=\lfloor x\rfloor =\frac{1}{2 \pi i} \int_{c-i \infty }^{c+i \infty } \frac{\zeta (s)}{s} x^s \, ds$$
$$\tag{1.2}g(s)=\frac{\zeta (s)}{s}=\int_0^{\infty } \lfloor x\rfloor x^{-s-1} \, dx$$
Functions $f(x)=x-\lfloor x\rfloor$ and $g(s)=\frac{-\zeta (s)}{s}$ are related by Mellin inversion theorem, for $0<c<1$, $0<\Re(s)<1$. $$\mathcal{M}_x(f(x))(s)=\mathcal{M}_s^{-1}(g(s))(x)$$
$$\tag{2.1}f(x)=x-\lfloor x\rfloor =\frac{1}{2 \pi i} \int_{c-i \infty }^{c+i \infty } \frac{-\zeta (s)}{s} x^s \, ds$$
$$\tag{2.2}g(s)=\frac{-\zeta (s)}{s}=\int_0^{\infty } (x-\lfloor x\rfloor) x^{-s-1} \, dx$$
Relations in $(1.1)$ and $(1.2)$ can be derived using Abel's summation formula, as is described on that Wikipedia page.
How relations $(2.1)$ and $(2.2)$ can be derived?
It is interesting that $(1.1)$ and $(1.2)$ are valid for $\Re(s)>1$ while $(2.1)$ and $(2.2)$ for $0<\Re(s)<1$.
Are there any other similar formulas that involve $\lfloor x\rfloor$, $\zeta (s)$ and Mellin inversion theorem that are valid at least in some portion of $\Re(s)<0$.
For $\Re(s)\in (-1,0)$ we have $$\zeta(s)=s\int_0^\infty (\lfloor x\rfloor-x+1/2) x^{-s-1}dx$$ whence for $\sigma\in (-1,0)$ $$\lfloor x\rfloor-x+1/2 = \lim_{k \to \infty}\frac1{2i\pi} \int_{(\sigma)} \frac{\zeta(s)}{s} x^s e^{s^2/k} ds\tag{1}$$ By the Cauchy integral theorem and the polynomial growth of $\zeta(s)$ on vertical strips, the RHS integrals stay the same for all $\sigma < 0$, so that $(1)$ stays valid for all $\sigma < 0$.
That is to say we have the inverse Fourier/Mellin transform in the sense of distributions $$\forall \sigma < 0, \qquad \mathcal{M}^{-1}\left[\frac{\zeta(s)}s\right]_{\Re(s)=\sigma}= \lfloor x\rfloor-x+1/2$$