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.



Is $\limsup_{x\to\infty}\big(\sum\limits_{d|3^x-1}{1/d}\big)/\big(\sum\limits_{p<x}1/p\big)<\infty$?
The OP's first display is true. Indeed, [Erdős (1971)](https://users.renyi.hu/%7Ep_erdos/1971-14.pdf) proved that
$$\sum\_{d\mid 2^x-1}\frac{1}{d}\ll\log\log x\ll\sum\_{p<x}\frac{1}{p},$$
and he noted that the proof works equally well for the divisors of $a^x-1$.


