This article is cited in
2 papers
A limit theorem for the Riemann Zeta-function close to the critical line. II
A. P. Laurincikas
Abstract:
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.
Bibliography: 8 titles
UDC:
511 +
519.2
MSC: Primary
11M06; Secondary
11M26,
11M41 Received: 04.07.1987 and 22.02.1989
Fulltext:
PDF file (615 kB)
