A limit theorem for the Riemann zeta-function close to the critical line

It is shown that as $T\to\infty$ the distribution function
$$ \frac1T\operatorname{mes}\{t\in[0,T],\ |\zeta(\sigma_T+it)|^\frac{1}{\sqrt{2^{-1}\ln\ln T}}<x\} $$
approaches the distribution function of the logarithmic normal distribution. Here $\operatorname{mes}\{A\}$ is the Lebesgue measure of the set $A$, and
$$ \sigma_T=\frac12+\frac{\sqrt{\ln\ln T}\psi(T)}{\ln T}, $$
where $\psi(T)\to\infty$ and $\ln\psi(T)=o(\ln\ln T)$ as $T\to\infty$.
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