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

We prove a central limit theorem for log vertical bar zeta(1/2 + it)vertical bar with respect to the measure vertical bar zeta((m)) (1/2 + it)vertical bar(2k) dt (k, m is an element of 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 vertical bar zeta(1/2 + it + i alpha)vertical bar(2k) dt, with alpha is an element of (-1, 1). Finally, we prove unconditionally the analogue result in the random matrix theory context.