So it is known that $$ \prod_{p\, \text{prime}}\frac{p}{p-1} = \sum_{k=1}^\infty \frac{1}{k}. $$
It is also known that for any particular harmonic number $H_n$, that we can bound it by $$ H_n = \sum_{k=1}^n \frac{1}{k} < \log(n)+1. $$
I was wondering if there was any similar bound for the above product of the first $m$ primes, hopefully in terms of some logarithm of $m$ $$ \prod_{k=1}^m\frac{p_k}{p_k-1} < f(m) $$
Just by performing some plotting in matlab, it appears that $f(m)=2(\log(m)+1)$ works, however I do not know if this (or a tighter bound) has been proven
If anyone could point me along the correct direction, I am having trouble because I cannot seem to find the name of this prime product, so finding resources is tough. Thank you