On the representation $\Gamma(z)=\sum_{n=2}^\infty \int_0^\infty a(n)e^{-e^{nx}}e^{nzx}dx$, for arithmetic functions $a(n)$ and $\Re z>0$

Question

Combining (see below the motivation where I explain what were the calculations) the integral representation for the Gamma function with the Prime Number Theorem I can deduce for each fixed complex number with $\Re z>0$ that $$\Gamma(z)=\sum_{n=2}^\infty \mu(n)\int_0^\infty e^{-e^{nx}}e^{nzx}dx,\tag{1}$$ where $\mu(n)$ is the Möbius function. I would like to interchage the sign of the integral and summation. I know that I need to combine with Fubini's theorem yo state that $$\int_0^\infty\sum_{n=2}^\infty\left|\mu(n) e^{-e^{nx}}e^{nzx}dx\right|<\infty,$$ thus I need to solve the following question.

Question 1. How do you justify that for each fixed complex number such that $\Re z>0$, and real numbers $x>0$ $$\sum_{n=2}^\infty \frac{e^{nx\Re z}}{e^{e^{nx}}}dx$$ is finite?

Motivation of previous question. I did the change of variable $t=e^{nx}$, where $n\geq 2$ is an integer, in the integral representation for the Gamma function $\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}dt$ to get using Riemann's trick with the multiplication by the Möbius function $(1)$, since $\sum_{n=2}^\infty\frac{\mu(n)}{n}=-1$, it is using the Prime Number Theorem.

Motivation to ask next question. Then Liouvile function $\lambda(n)$ also shuould be satisfies a similar statement, $$\Gamma(z)=\sum_{n=2}^\infty \int_0^\infty \lambda(n)e^{-e^{nx}}e^{nzx}dx$$ if we presume that we can interchange the signs of sum and the integration (that is, if the statement in Question 1 holds) because Liouvile function satisfies a Prime Number Theorem, and $\lambda(1)=1$.

Question 2. Can you provide us a characterization (in terms of a criterion*) for arithmetic function $a(n)$ such that $$\Gamma(z)=\sum_{n=2}^\infty \int_0^\infty a(n)\frac{e^{nzx}}{e^{e^{nx}}}dx\tag{2}$$ holds for $\Re z>0$?

Remark. The remark* is due that isn't required find all arithmetic functions satisfying previous representation $(2)$, what I am asking with the word characterization is a criterion in terms of, maybe if $a(n)$ is a completely multiplicative function, or maybe the criterior is about its average order...I don't know. I don't know what criterion is feasible with the purpose to state a characterization of those $a(n)$ satisfying $(2)$. Many thanks.

Answer 1

It looks to me that there is an unnecessary "twist" in your integrals. We have $$ \int_{0}^{+\infty} a(n) \frac{e^{nzx}}{e^{e^{nx}}}\,dx \stackrel{x\mapsto \log t}{=} \int_{1}^{+\infty}a(n)\frac{t^{nz-1}}{e^{t^n}}\,dt \stackrel{t\mapsto u^{1/n}}{=} \int_{1}^{+\infty}a(n)\,u^{z-1} e^{-u}\,du$$ and $\int_{1}^{+\infty}u^{z-1}e^{-u}\,du = \Gamma(z,1)$ does not depend on $n$, so your Question 2 is asking for sequences such that $\frac{\Gamma(z)}{\Gamma(z,1)}=\sum_{n\geq 2}a(n) $ holds for every $z$ in the right half-plane: none, obviously.

And your Question 1 is trivial: $$\sum_{n\geq 2}\exp\left(\gamma n x-e^{nx}\right) $$ is convergent for any $\gamma>0$ since its terms decay really fast to zero.