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

We prove a central limit theorem for $\log|ζ(1/2+it)|$ with respect to the measure $|ζ^{(m)}(1/2+it)|^{2k}dt$ ($k,m\in\mathbb N$), 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 $|ζ(1/2+it+iα)|^{2k}dt$, with $α\in(-1,1)$. Finally we prove unconditionally the analogue result in the random matrix theory context.

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