What is the best known lower bound for $$\sum_{p \le x} \frac{1}{p}?$$ The best one I found was $$\sum_{p \le x} \frac{1}{p} \ge B + \ln{\ln{x}} - \left(\frac{1}{10\ln ^2 x} + \frac{4}{15\ln ^3 x}\right),$$ extracted from Pierre Dusart's "Sharper bounds for $\psi, \theta, \pi, p_k$."
Has anyone made any improvement on it since then?
Thank you!