I am currently trying to understand the proof of the simplest zero-density theorem for $\zeta$ there is, namely $$N(T+1)-N(T-1) \ll \log(T),$$ where $T>2$ and $N(T) = \#\{\rho \in \mathcal N : |\Im(\rho)| < T \}$ with $\mathcal N$ the set of non-trivial zeroes of $\zeta$. I consulted three books on analytic number theory so far (by Brüdern, Vaugan and Tenenbaum) and in all three of those the proof is (more or less) the same: We use Jensen's Formula to obtain $$\int_0^1 \log | \zeta(2+iT+re(\theta))| d\theta = \log|\zeta(2+iT)| + \sum_{\rho \in \mathcal N \text{ and } |\rho - (2+iT)| < r} \log \left(\frac r {|\rho-(2+iT)|} \right)$$ for some $r \in [3,4]$ and $e(\theta) = \text{e}^{2\pi i \theta}$. In this formula, it is relatively easy to see that the sum on the right hand side is $\approx N(T+1)-N(T-1)$. So far so good. It is now stated that a bound of the form $$\zeta(\sigma + it) \ll t^k$$ for $t>1$, $\sigma \in \mathbb R$ and some $k$ of fitting size implies the bound $$\log|\zeta(s)| \ll \log |t|.$$ Tenenbaum even explicitly mentions the bound $$\log |\xi(s)| \ll |s|\log|s| \text{ (as |s| }\rightarrow \infty \text{ for }\Re s \geq \tfrac 12) $$ for the complete Riemann-Zeta function in his book Introduction to analytic and probabilistic number theory as Formula (41) on page 152. It is clear that these bounds imply the desired result.
I could agree with the authors if they stated (and used) these bounds as $\log |\zeta(s)| \leq O(\log |s|)$, but in general these bounds are clearly wrong (as long as I've not completely lost it)! $\zeta$ (and $\xi$) have zeroes with arbitrarily large imaginary part, and $\log |\zeta|$ ($\log |\xi|$ resp.) has to be large near those.
Am I missing something? It might well be that I am misinterpreting some notations here, it's hard to believe that all three (or even one) of these authors would have published this without thinking about my issue.