I don't know if this set of calculations is right o well known. I was inspired in the calculations from the Proof (in Démonstration do a click in [afficher] to see the well known proof) of section 2.5 Expression intégrale from the French article of the Wikipedia for the Riemann Zeta function.
Question. Have mathematical meaning my calculations? Are there mistakes? Do you know if is defined in the literature $$\sum_{k=1}^\infty\frac{\mu(k)}{\left(e^{nt}\right)^k}$$ (here $n\geq 1$ is a fixed integer) for $t>0$?
For each integer $k\leq 1$ we multiply by $\frac{\mu(k)}{k^s}$ such integral expression and take the sum from $k=1$ to infnite to get for $\Re s>1$ (since the Riemann zeta function dosen't vanish in this half-plane) $$\Gamma(s)=\sum_{n=1}^\infty\sum_{k=1}^\infty\mu(k)\int_0^\infty e^{-u}\left(\frac{u}{kn}\right)^{s-1}\frac{du}{kn},$$ then using the change of variable $u=knt$ by the Dominated Convergence Theorem we've $$\Gamma(s)=\sum_{n=1}^\infty\int_0^\infty g(n,t)t^{s-1}dt$$ where thus we define $$g(n,t):=\sum_{k=1}^\infty\frac{\mu(k)}{\left(e^{nt}\right)^k}$$ (for each $n\geq 1$ and $t>0$). And here seeing my definition I don't know if converges, but by comparison with previous integral expression, if there are no mistakes in my calculations I believe that does converge, also one has the comparison with the integral representation for the Gamma function yielding thus $$\sum_{n=1}^\infty g(n,t)=e^{-t}$$ for reals $t>0$.
I don't know if I had some great mistake, and also I've tried relate previous series using Möbius inversion for the Taylor series of the logarithm, but I don't know how get it. Where were my mistakes? Many thanks.