In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia *Dedekind psi function*. On the other hand I add the reference that Wikipedia has the article *Fermat number*, $F_l=2^{2^l}+1$ and that I was inspired in the results showed in page 101 of [1].

The Dedekind psi function can be represented for a positive integer $m>1$ as $$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$$ with the definition $\psi(1)=1$.

Question.I would like (what work can be done about it) if one can to deduce some claim about the behaviour of $$\frac{\psi(F_m)}{F_m}$$ as $m\to \infty.$Many thanks.

If the question is in the literature please answer the question as a reference request and I try to search and read the results from the literature. If the question is very difficult I ask about the behaviour or heuristic for the quotient $\frac{\psi(F_m)}{F_m}$ for very large integers $m\geq 1$.

## References:

[1] Michal Krizek, Florian Luca, and Lawrence Somer, *17 Lectures on Fermat Numbers*, CMS Books in Mathematics, Springer (2001).

4430576asked two months ago. I've edited the question and few minutes after I got a downvote in Mathematics Stack Exchange. The guy who downvote the question isn't interested neither in respect the ownship of the post (that is the site MSE) nor in the question. Thus I'm going to ask the question here and delete the post from MSE. (My apologizes for this polemics, but it seems to me unfair) $\endgroup$