On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$\sum_{n=1}^\infty\frac{p_n^{1/n}-p_{n+1}^{1/(1+n)}}{n^s}<2+\left(\frac{1}{2^s}-1\right)3^{\frac{1}{2}}+\left(\frac{1}{3^s}-\frac{1}{2^s}\right)5^{\frac{1}{3}}+\left(\frac{1}{4^s}-\frac{1}{3^s}\right)7^{\frac{1}{4}}+\ldots$$ that is a direct consequence of the conjecture.
It was the only calculation that this morning did to explore the conjecture using Dirichlet series. My purpose is learn easy facts on Dirichlet series, and edit funny questions in this site. I am specially interesting in the calculation of abscissas of absolute convergence because those are the keys to start the calculations for bounds of partial sums of Dirichlet series $|\sum^N|$. Then
Question. It is possible to determine approximately the abscissa of absolute convergence of previous Dirichlet series? Thanks in advance.
My calculations were that there exists the corresponding abscissas of convergence absolute for the Dirichlet series $\sum_{n=1}^\infty\frac{p_n^{1/n}}{n^s}$ and $\sum_{n=1}^\infty\frac{p_{n+1}^{1/(n+1)}}{n^s}$, say us $\sigma_a$ and $\sigma_{a'}$ respectively. I've deduced this since (and there is a similar calculation for the other Dirichlet series, the first of them) the Prime Number Therorem implies $$p_{n+1}^{\frac{1}{n+1}}\sim ((n+1)\log(n+1))^{\frac{1}{n+1}}$$ thus $\sim 1$ since $$\lim_{n\to\infty}\frac{1}{n+1}\log((n+1)\log(n+1))=0,$$ and from this I've said that there exists a real $\sigma_{a'}$ being the abscissa of absolute convergence of $$\sum_{n=1}^\infty\frac{p_{n+1}^{1/(n+1)}}{n^s}.$$
Thus I believe that it is neccesary an approximation for $\sigma_{a}$ and $\sigma_{a'}$ with the purpose to claim that $\max\{\sigma_{a}, \sigma_{a'}\}$ is the required abscissa of absolute convergence that answers the Question.
I don't know if this exercise was in the literature. But there are a lot of literature in arXiv. If there are some useful fact from those, it is you recognize that some recipe that could be useful, add the reference and I will try read the proposition in arXiv.