On the Logarithm of the Riemann Zeta-function Near the Nontrivial Zeros

Abstract

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