Let $s$ range over the complex numbers and write such numbers as $s=\sigma+iT$ with $\sigma,T$ real. In textbooks on analytic number theory, I have found the following estimate: $$\frac{\log|s|}{|s|} \ll \frac{\log T}{T}$$ for all $\sigma \leq -1$ and all $T\geq 2$ where the implicit constant is independent of $\sigma$ and $T$. How to rigorously prove this?
The estimate above is used in the proof of the explicit formula in the textbook on analytic number theory by Davenport as well as in the book by Montgomery and Vaughan.