I know from Dusart papers and other similar papers that, $ c_1 + \ln \theta(x) \leq \sum \limits_{p \leq x} \frac{\ln p}{p} $ and that $c_0+ \ln \ln \theta(x) \leq \sum \limits_{p \leq x} \frac{1}{p} $ are equivalent to R.H. ($c_0,c_1$ are known and explicit values but are not important to the question here)
Recently I came across a formula that relates $\theta(x)$ to summations like $ \sum \limits_{p \leq x} \frac{\ln^2 p}{p}$ and $ \sum \limits_{p \leq x} \frac{\ln^3 p}{p}$, so my question is : is the R.H. equivalent to $ c_2 + \frac{\ln^2 \theta(x)}{2} \leq \sum \limits_{p \leq x} \frac{\ln^2 p}{p}$ ??
If the above is true, then can we always build an inequality for a polynomial in respect to $\ln p$ such as $ \ln^3 p- 3 \ln^3 p +7 \ln p -11$ in the summation that is equivalent to R.H. ?
A ref to published paper is good answer too.
Note : for constant polynomial or linear polynomial with respect to $\ln p$ the answer is true, simply be applying robin's inequality multiplied with constant.
Thanks.