I just read in Martin Klazar's Intoduction to Number Theory (page 53), that $\sum_{p\leq x} \log p - \log (p-1) = \log\log x + \gamma + O(1/\log x)$.
Where $\gamma$ is the Euler-Mascheroni constant, $p$ is a prime,
and the error term $O(1/\log x)$ was evaluated from the integral $\int_x^{\infty} \frac{z(t)}{t\log^{2}t} \mathrm {d}t$ and $z(t) = O(1)$.
My question is on this error term: is it nonnegative such that we have
$\sum_{p\leq x} (\log p - \log (p-1)) > \log\log x + \gamma $ for sufficiently large $x$ ?
Or is there any other means by which we can deduce (or refute) this inequality ?