Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\right|\leq1}\frac{1}{s-\rho}+O\left(\log\left(t\right)\right) \tag{1}$$ uniformly for $-1\leq\sigma\leq2$ (see for example Titchmarsh, “The theory of the Riemann zeta function”, second ed., page $217$). So in particular if we take $\sigma=0$ holds $$\frac{\zeta'}{\zeta}\left(it\right)=\sum_{\left|t-\gamma\right|\leq1}\frac{1}{it-\beta-i\gamma}+O\left(\log\left(t\right)\right).
$$ **Question**: is it possible to prove that $$ \sum_{\left|t-\gamma\right|\leq1}\frac{1}{it-\beta-i\gamma}\ll\log\left(t\right)?
$$ The problem is that it could be some zeros with real part very close to $0$ and so for $t=\gamma$ the sum is very "big". Thank you.

**Addendum**: My final goal it's to prove that $$\frac{\zeta'}{\zeta}\left(it\right)=O\left(\log\left(t\right)\right)$$ so also another proof of it (if exists) without the use of $(1)$ is welcome.