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

### Maksym Radziwiłł

McGill University, Montreal, Canada### Kannan Soundararajan

Stanford University, USA

## Abstract

We present a new and simple proof of Selberg's central limit theorem, according to which the logarithm of the Riemann zeta-function at height $t$ is approximately normally distributed with mean 0 and variance $\frac {1}{2}$ log log $t$.

## Cite this article

Maksym Radziwiłł, Kannan Soundararajan, Selberg’s central limit theorem for log $|\zeta(\tfrac 12+it)|$. Enseign. Math. 63 (2017), no. 1/2, pp. 1–19

DOI 10.4171/LEM/63-1/2-1