What is known about the asymptotic behavior of
$$
f(x)=\sum\_{p\le x}\frac1p-\log\log x-B\_1?
$$


Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega\_\pm$, say)? A reference would be great.



Error term for prime harmonic
["Mertens' Proof of Mertens' Theorem"](http://arxiv.org/abs/math/0504289) suggests that Mertens had an error term of $O\left(\frac1{\ln x}\right)$, though that's not tight; theorem 14 there offers an $O\left(\frac1{\ln^2x}\right)$ unconditional estimate and an $O(x^{-1/2}\ln x)$ estimate dependent on RH, with references to the original proofs.


