In essence, I need to determine when a number is within this set: $$\Big\{ t : t \in \mathbb{R} \;\wedge\; \Big|1+ i\xi'\big(\frac12 + i t\big)\Big| < 1 \Big\},$$ where $\xi'$ is the first derivative of the Riemann xi function. This is assuming RH so it's just the critical strip.
Is there a way of simplifying this equation into something more manageable? Or is there a better way of determining this altogether?
On $|\Re(s)-1/2|\le 1/4, |\Im(s)|> 1$
$|\zeta(s)|\le 2|s|$ this follows from $\zeta(s)=\frac{s}{s-1}+s\int_1^\infty(\lfloor x\rfloor-x)x^{-s-1}dx$.
$\Gamma(s/2)$ is rapidly decreasing, this is because $|\Gamma(s/2)\prod_{k=0}^{K-1}(s/2+k)|\le \Gamma(1/2+K)$
From there you get some $C=16$ such that $|\xi(s)| \le C /|s|$. This implies by the Cauchy integral formula that $|\xi'(s)|\le 32 C/|s|$.
$\xi(1/2+it)$ is real: this follows from the functional equation that $\Gamma(s/2)\pi^{-s/2}\zeta(s)$ is invariant under $s\to 1-s$ and is complex conjugated under $s\to \overline{s}$, thus $i\xi'(1/2+it)$ is real.
$\xi(1/2+it)$ changes of sign infinitely often (this is a bit less obvious) thus $i\xi'(1/2+it)$ changes of sign infinitely often. Assuming that there are no double zeros, exactly half of non-trivial zeros $1/2+it, t> 32C$ are attractive fixed points of $s\to s+\xi(s)$. Assuming the RH there are no others in the critical strip.