Selberg's central limit theorem for $\log |\zeta(\tfrac 12+it)|$

TL;DR

This paper offers a new, simplified proof of Selberg's central limit theorem, demonstrating that the logarithm of the Riemann zeta function's absolute value on the critical line follows a normal distribution with specific mean and variance.

Contribution

The paper introduces a novel, streamlined proof of Selberg's central limit theorem for the Riemann zeta function.

Findings

Log |zeta(1/2 + it)| is approximately normally distributed

Mean of the distribution is 0

Variance of the distribution is (1/2) log log t

Abstract

We present a new and simple proof of Selberg's central limit theorem, according to which $\log |\zeta(\tfrac 12 + it)|$ is approximately normally distributed with mean $0$ and variance $\tfrac 12 \log\log t$.