I would like to know if the following comparison, of series involving sequences arising from analytic number theory and the so-called Carleman's inequality, is potentitally interesting. Thus I am asking it as a sof question or well to know what about it, or if this idea was in the literature.
One has the Wikipedia article dedicated to the Carleman's inequality, is this. On the other hand one has sequences $a_n$ of positive integers or non-negative real numbers from analytic number theory. For example $\frac{2+\mu(n)}{n^{x}}$ where $\mu(n)$ is the Möbius function and $x>1$, other example is $|G_n|$ where $G_n$ is the $n$th Gregory coefficient (see this Wikipedia), or finally this example of $a_n=\frac{1}{p_n}+\frac{1}{2+p_n}$, where $p_n$ is prime and $p_n+2$ also is prime.
Question. Imagine that a friend asks me if is interesting to write the specializations of Carleman's inequality that I've evoked. That is, if the resulting inequalities have a good mathematical meaning: are sharper inequalitites (are good, or very difficult to improve), or are showing some interesting fact about some of our sequence $a_n$. What should I say to him/her?* Many thanks.
*If you argue that this inequality might not be very good for some of my sequences, or the calculations are tedious, please explain your words in your answer of the previous question.
Also you can to illustrate your words with different sequences from analytic number theory.