Vanishing property of logarithmic derivative of zeta function

Question

I was trying to derive the explicit formula for the integrated Chebyshev $\psi$ function, $\psi_1$ defined as

\begin{equation}\psi_1(x)=\int_1^x\psi(y)dy\end{equation}

But I have stumbled upon one technicality, and that is showing that the quantity \begin{equation} \frac{\zeta'(s)}{\zeta(s)}\frac{x^{s+1}}{s(s+1)} \end{equation} vanishes as $|s|\to\infty$. I have a hunch that the logarithmic derivative of the zeta function vanishes as $|s|\to\infty$, but is it really true? If so, how?

Answer 1

Hint: $~\displaystyle\lim_{n\to\infty}2^n\cdot\zeta'(n)=-\ln2.~$ The rest is trivial.