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

A. P. Laurincikas

Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 177-193

Résumé

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 $$ \frac1T\operatorname{mes}\{t\in[0,T],\ |\zeta(\sigma_T+it)|(2^{-1}\ln\Delta_T)^{-1/2}\}, $$ 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. Bibliography: 8 titles