This question is related to the non-trivial zeta zero counting function defined in (1) below and its asymptotic defined in (2) below.
(1) $\quad N(t)=\sum\limits_{0<\Im(\rho)\le t}1$
(2) $\quad\overset{\text{~}}{N}(t)=\frac{t}{2 \pi}\left(\log\left(\frac{t}{2 \pi }\right)-1\right)+\frac{\log(2 \pi)}{2},\quad N(t)\approx\overset{\text{~}}{N}(t)$
Now consider the averaging function defined in (3) below.
(3) $\quad g(K)=\frac{1}{K}\sum\limits_{k=1}^K\left(N\left(\Im\left(\rho_k\right)\right)-\overset{\text{~}}{N}\left(\Im\left(\rho _k\right)\right)\right)$
The following figure illustrates a discrete plot of the function $g(K)$ defined in formula (3) above in blue. The horizontal orange line represents $\frac{\log(2 \pi)}{4}$ which was my initial guess for $\underset{K\to\infty}{\text{lim}}g(K)$ but the figure illustrates this guess is not quite correct.
Figure (1): Illustration of discrete plot of $g(K)$ defined in formula (3) above (blue)
Question: Does $\underset{K\to\infty}{\text{lim}}g(K)$ have a closed form representation? If not, what is the most accurate known estimate for $\underset{K\to\infty}{\text{lim}}g(K)$?
After posting this question I noticed the following corollary from Montgomery and Vaughn which assumes $N(t)=(N(t^+)+N(t^-))/2$ where $N(t)=\sum\limits_{0<\Im(\rho)<t}1$ and $s(t)=\frac{1}{\pi}\arg\left(\zeta\left(\frac{1}{2}+i t\right)\right)$.
(4) $\quad N(t)=\frac{t}{2 \pi}\left(\log\left(\frac{t}{2 \pi}\right)-1\right)+\frac{7}{8}+s(t)+O\left(\frac{1}{t}\right),\quad t\geq 2$
This suggests formula (5) below provides a better asymptotic for $N(t)$ defined in formula (1) above than formula (2) above.
(5) $\quad\overset{\text{~}}{N}(t)=\frac{t}{2 \pi}\left(\log\left(\frac{t}{2 \pi }\right)-1\right)+\frac{11}{8},\quad N(t)\approx\overset{\text{~}}{N}(t)$
The following figure illustrates a discrete plot of the function $g(K)$ defined in formula (3) above using the definitions of $N(t)$ and $\overset{\text{~}}{N}(t)$ defined in formulas (1) and (5) above.
Figure (2): Discrete plot of $g(K)$ from formula (3) using formulas (1) and (5) for $N(t)$ and $\overset{\text{~}}{N}(t)$
Question: Is it true $\underset{K\to\infty}{\text{lim}}g(K)=0$ assuming the definitions of $N(t)$ and $\overset{\text{~}}{N}(t)$ in formulas (1) and (5) above?
Now consider the $f(n)$ and $a(n)$ functions defined in formulas (6) and (7) below which assume the definitions of $N(t)$ and $\overset{\text{~}}{N}(t)$ in formulas (1) and (5) above and where $g(K)$ illustrated in Figure (2) above corresponds to $g(K)=\frac{1}{K}\sum\limits_{n=1}^K f(n)$.
(6) $\quad f(n)=N\left(\Im\left(\rho_n\right)\right)-\overset{\text{~}}{N}\left(\Im\left(\rho_n\right)\right)$
(7) $\quad a(n)= \begin{array}{cc} \{ & \begin{array}{cc} 1 & f(n)>0 \\ 0 & \text{True} \\ \end{array} \\ \end{array}$
The first $28$ values of $a(n)$ defined in formula (7) above match the first $28$ values of OEIS entry A100060, but if it's true $\underset{K\to\infty}{\text{lim}}g(K)=0$ then it seems to me $a(n)$ defined in formula (7) above provides more precise information than OEIS entry A100060.