On the logarithm of the Riemann zeta-function near the nontrivial zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate distribution of sequences $(\log|\zeta(\rho+z)|)$ $(\arg\zeta(\rho+z)).$ Here $\rho=\frac12+i\gamma$ runs over nontrivial zeros zeta-function, $0<\gamma \leq T,$ $T$ is a large real number, $z=u+iv$ nonzero complex number modulus $\ll 1/\log T.$ Our approach proceeds via study integral moments these sequences. If let $z$ tend to $0$ further assume that all $\rho$ are simple, can replace pair correlation conjecture with weaker spacing on deduce sequence $(\log (|\zeta^\prime(\rho)|/\log T))$ has an approximate Gaussian mean variance $\frac12\log\log This gives alternative proof old result Hejhal improves it by providing rate convergence distribution.