A limit theorem for the Riemann Zeta-function close to the critical line. II

Let $\Delta_T\to\infty$, $\Delta_T\leq\ln T$, and $\psi_T\to\infty,\ \ln\psi_T=o(\ln\Delta_T)$, as $T\to\infty$, and let $\displaystyle\sigma_T=\frac12+\frac{\psi_T\sqrt{\ln\Delta_T}}{\Delta_T}$. In this paper we study the asymptotic behavior of the Riemann $\zeta$-function on the vertical lines $\sigma_T+it$. We prove that the distribution function
$$ \frac1T\operatorname{mes}\{t\in[0,T],\ |\zeta(\sigma_T+it)|(2^{-1}\ln\Delta_T)^{-1/2}<x\}, $$
converges to a logarithmic normal law distribution function as $T\to\infty$, and that, if $\exp\{\Delta_T\}\leqslant(\ln T)^{\frac23}$, then the measure
$$ \frac1T\operatorname{mes}\{t\in[0,T],\ \zeta(\sigma_T+it)(2^{-1}\ln\Delta_T)^{-1/2}\in A\}, \quad A\in\mathscr B(C), $$
is weakly convergent to a nonsingular measure.
The proof of the first assertion uses the method of moments, and that of the second uses the method of characteristic transformations.
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