TITLE: Is $\limsup_{x\to\infty}\big(\sum\limits_{d|3^x-1}{1/d}\big)/\big(\sum\limits_{p<x}1/p\big)<\infty$?
QUESTION [12 upvotes]: Can anyone provide any suggestion if the following is true for natural number $x$?
$$\limsup_{x\to\infty}\dfrac{\sum\limits_{d|3^x-1}{\frac{1}{d}}}{\sum\limits_{p<x}\dfrac{1}{p}}<\infty$$ where $p$ runs over primes.
P.S I know that
the bound ${\sum\limits_{d|n}{\frac{1}{d}}}<e^{\gamma}\log\log n +O(1)$is true so maybe it could help.

REPLY [4 votes]: One remark for the OP. On certain well-believed conjectures, the exact value of the limsup can be determined. The answer involves $\gamma$, $\log 3$, and an integral expression with Dickman's rho function. See this arXiv paper: https://arxiv.org/abs/2006.02373 (Theorem 1.2 and section 6)