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

TL;DR

Under the assumptions of the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, the paper studies the distribution of the logarithm and argument of the zeta-function near its zeros, providing new insights and an improved proof of a classical result.

Contribution

The paper introduces a new approach to analyze the distribution of the logarithm of the zeta-function near zeros, relaxing the pair correlation conjecture to a weaker hypothesis and offering an improved rate of convergence.

Findings

The sequence $( ext{log}| ext{ extzeta}( ho+z)|)$ is studied under RH and pair correlation conjecture.

The distribution of $( ext{log}| extzeta'( ho)|/ ext{log} T)$ is approximately Gaussian with mean 0 and variance (1/2)log log T.

An alternative proof of Hejhal's result with a rate of convergence is provided.

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.