Inspired by the OP question 'Why are they distributed as such?' I took another look at the paper of Hejhal on the real part of $\log\zeta^\prime$ cited in the previous answer, and (after 30 years) I finally understand a little. Here's my exposition of Hejhal's exposition (p. 346) of the basic idea behind the proof. First some notation:

With $\chi(s)=\pi^{s-1/2}\frac{\Gamma((1-s)/2)}{\Gamma(s/2)}$, define $\phi(s)$ by
$\chi(s)^{-1/2}=\exp(i\phi(s))$, so $\phi$ is real on the critical line.

Let $M$ be a large constant, $t$ an auxiliary random variable with $T\le t\le 2T$, and $W_t$ the 'window' $[t-M/\log T,t+M/\log T]$.

Let $A(t)=\frac{t}{2\pi}\log\left(\frac{t}{2\pi}\right)-\frac{t}{2\pi}$.

Let $x=A(u)$, and $\theta(u)=\phi(1/2+iu)$. Let $P_t(x)$ be the polynomial approximation $\Pi_{\gamma\in W_t}(x-A(\gamma))$, and define $\Omega_t(u)$ to be the correction to the approximation so that

$
\zeta(1/2+iu)=\exp(\Omega_t(u)-i\theta(u))P_t(x).
$

Computing logarithmic derivative (in $u$, being careful with the chain rule) we see

$
i\zeta^\prime(1/2+iu)/\zeta(1/2+iu)=\Omega_t^\prime(u)-i\theta^\prime(u)+P_t^\prime(x)/P_t(x)\cdot A^\prime(u)
$

Rearranging gives

$
i\zeta^\prime(1/2+iu)/A^\prime(u)=\zeta(1/2+iu)\left(\Omega_t^\prime(u)/A^\prime(u)-i\theta^\prime(u)/A^\prime(u)+P_t^\prime(x)/P_t(x)\right).
$

Hejhal will make an estimate (see below) of the term in parenthesis on
the right to argue that

$\log\left|\zeta(1/2+iu)\right|/\sqrt{\log\log T}$ and
$\log\left|\zeta^\prime(1/2+iu)/A^\prime(u)\right|/\sqrt{\log\log T}$
are (in effect) the same random variable, and so Selberg's theorem
(mentioned in the previous answer) applies.

For this estimate, Hejhal claims he and Bombieri showed previously that the total variation of $\Omega_t(u)$ on $W_t$ is $O_M(1)$ for 'most' $t$. From this,

$
\int_{W_t}\left|\Omega^\prime_t(u)\right|du=O_M(1)
$
for 'most' $t$, and so on 'most' windows $W_t$, $\left|\Omega^\prime_t(u)\right|=O_M(\log T)$.

The above was the hard part; $\theta^\prime(u)/A^\prime(u)$ is elementary. And $P_t^\prime(x)/P_t(x)=\sum_{\gamma\in W_t}1/(x-A(\gamma)),$ with average spacing between $A(\gamma)$ being 1 and the number of terms in the sum $O_M(1)$. Hejhal argues heuristically that

$
\log\left|\Omega_t^\prime(u)/A^\prime(u)-i\theta^\prime(u)/A^\prime(u)+P_t^\prime(x)/P_t(x)\right|=O_M(1)
$
except for a subset of small measure.

From this one see the difficulty in extending this to the imaginary part of $\log\zeta^\prime(1/2+iu)$, defined (say) by continuous variation up the vertical line from $4$ to $4+iu$ and along the horizontal line from $4+iu$ to $1/2+iu$. Namely, the estimates depend on $u$ being inside the window $W_t$.

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