Given $P_k$, the truncated Prime $\zeta$ function, defined like $$ P_k(it):=\sum_{n=1}^k p_n^{it}, $$ where $p_n$ is the $n$th prime. What's the probability to find a value or range of $t$ less than $T$, where $|P_k(it)|<\epsilon$?
EDIT By that I mean, among all values of $t<T$, how large is the portion of values, where $|P_k(it)|<\epsilon$?
To give an example, I plotted $P_{500}(it)$, where $0<t<400$:
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The lower plot shows the portion of values that lies below a certain threshold. Almost all values lie below $.1$. I interpret the lower plot as the sum over the distribution.