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

Abstract

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.

17 pages