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

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences (log|ζ(ρ+ z)|) and (argζ(ρ + z)). Here ρ = 1 2 + iγ runs over the nontrivial zeros of the zeta-function, 0 < γ ≤ T, T is a large real number, and z = u + iv is a nonzero complex number of modulus ≪ 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 ρ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence (log(|ζ'(ρ)|/log T)) has an approximate Gaussian distribution with mean 0 and variance 1 2 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.