We define the Dirichlet density of a set $P$ of primes to be the limit $$\lim_{s \longrightarrow 1+} \frac{\sum_{p \leq x, p \in P} p^{-s}}{\log \left( \frac{1}{s-1} \right)}$$ whenever it exists and the logarithmic density of $P$ to be the limit $$\lim_{x \longrightarrow \infty} \frac{1}{\log \log x}\sum_{p \leq x} \frac{1}{p}$$ again subject to existence of the same. Of course these have corresponding "natural" analogues (as in there do exist notions of Analytic and Logarithmic Density over $\mathbb{N}$ and to the best of my knowledge, Tenenbaum proves that analytic density of a set $A \subset \mathbb{N}$ (over $\mathbb{N}$) exists if and only if the corresponding logarithmic density does, in which case the values of the two are equal).
I was curious about what happens in the above situation. Here (Dirichlet density) it has been proven that the existence of the above logarithmic density implies that of the above Dirichlet Density, and in that case the values of the two are equal. However, I have not been able to find a proof or counterexample for the opposite implication. Any help or suggestions would be really appreciated. Thanks a lot.