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

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 $\frac{1}{2}\log\log T$.