Weighted value distributions of the Riemann zeta function on the critical line

We prove a central limit theorem for $\log|\zeta(1/2+it)|$ with respect to the measure $|\zeta^{(m)}(1/2+it)|^{2k}dt$ ($k,m\in\mathbb N$), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure $|\zeta(1/2+it+i\alpha)|^{2k}dt$, with $\alpha\in(-1,1)$. Finally we prove unconditionally the analogue result in the random matrix theory context.