The distribution of the logarithmic derivative of the Riemann zeta-function

TL;DR

This paper studies how the logarithmic derivative of the Riemann zeta-function behaves near the critical line, showing it converges to a 2D Gaussian distribution with bounds on the convergence rate.

Contribution

It provides the first detailed analysis of the distribution of ta'/ta(s) near the critical line, including convergence rates to Gaussianity.

Findings

Distribution of ta'/ta(s) converges to a 2D Gaussian near the critical line

Established upper bounds on the convergence rate

Results enhance understanding of the zeta-function's behavior in critical regions

Abstract

We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=\sigma, where \sigma, lies in a certain range near the critical line \sigma=1/2. For such \sigma, we show that the distribution of \zeta'/\zeta(s) converges to a two-dimensional Gaussian distribution in the complex plane. Upper bounds on the rate of convergence to the Gaussian distribution are also obtained.