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|?(?+z)|) and (arg?(?+z)). Here ?=12+i? runs over the nontrivial zeros of the zeta-function, 0<??T, T is a large real number, and z=u+iv is a nonzero complex number of modulus ?1/logT. 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 ? are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence (log(|??(?)|/logT)) has an approximate Gaussian distribution with mean 0 and variance 12loglogT. This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.