Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. What is the best known upper bound for the order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $t\rightarrow\infty$? I am interested in answers both, assuming and without assuming the Riemann hypothesis. Can you please supply a reference? I am thinking that no upperbound is known since Spira [1] proves that the zeros of $\zeta’$ on the critical line, if they exist, necessarily coincide with zeros of $\zeta$. But the result by Spira is old.
[1] Spira, R., On the Riemann zeta function, Journal of the London Mathematical Society, 44 (1969), 325-328
