By Mertens theorem we know that $$ \sum_{p \leq x} \frac{\log p}{p} = \log x +O(1). $$
Is it possible to know something stronger of the type $$ \sum_{p \leq x} \frac{\log p}{p} = \log x +A +E(x), $$ where $A$ is some constant (and we don't care about the precise value of it) and $E(x)$ is some error term clearly better than $O(1)$?
Does a similar thing hold for a fixed arithmetic progression, say, $p \equiv 1 \pmod{4}$?