A weighted central limit theorem for $\log|\zeta(1/2+it)|$

TL;DR

This paper proves a weighted central limit theorem for the logarithm of the Riemann zeta function's absolute value, showing it is approximately normally distributed with specific mean and variance under the Riemann Hypothesis.

Contribution

It establishes a weighted CLT for log| extzeta(1/2+it)|, revealing its normal distribution behavior with respect to a specific measure, under the Riemann Hypothesis.

Findings

Distribution of log| extzeta(1/2+it)| is approximately normal

Mean of the distribution is log\u007d extlog T

Variance of the distribution is (1/2) extlog extlog T

Abstract

Under the Riemann Hypothesis, we show that as $t$ varies in $T\leq t \leq 2T$, the distribution of $\log|\zeta(1/2+it)|$ with respect to the measure $|\zeta(1/2+it)|^2dt$ is approximately normal with mean $\log\log T$ and variance $\frac{1}{2}\log\log T$.