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

Abstract

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

18 pages