This question is about statistical properties of the distribution of the complex part of non-trivial zeros $\rho_n$ of the Riemann $\zeta$‑function. The zeros tend to become more dense as $n$ grows, but we can compensate for that if we consider $\tau_n=\theta\left(\Im\,\rho_n\right)/\pi$ instead, were $\theta(x)$ is the Riemann–Siegel θ‑function. These values appear to lie close to the straight line: $\tau_n\approx n-3/2$, where the absolute error apparently does not exceed 1. Update: I've found slightly larger deviations among $2\cdot10^5<n<3\cdot10^5$.
Is this known to be true (at least, if we assume the Riemann hypothesis)?
If we consider $\tau_n-\left(n-3/2\right)$ as a random variable, can we determine its standard deviation and other statistical properties?
Do we know what distribution does it follow? It looks close to normal.
