Geodesic
A limit theorem for the Riemann Zeta-function close to the critical line. II
Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 177-193 
A. P. Laurincikas. A limit theorem for the Riemann Zeta-function close to the critical line. II. Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 177-193. http://geodesic.mathdoc.fr/item/SM_1990_67_1_a10/
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     title = {A limit theorem for the {Riemann} {Zeta-function} close to the critical {line.~II}},
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     url = {http://geodesic.mathdoc.fr/item/SM_1990_67_1_a10/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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

[1] Laurinchikas A. P., “Predelnaya teorema dlya dzeta-funktsii Rimana vblizi kriticheskoi pryamoi”, Matem. sb., 135(177) (1988), 3–11 | MR

[2] Laurinchikas A. P., “Predelnaya teorema dlya dzeta-funktsii Rimana na kriticheskoi pryamoi. I”, Lit. matem. sb., XXVIII:1 (1987), 113–132 | MR

[3] Gabriel R. M., “Some results conserning the integrals of moduli of regular functions along certain curves”, J. London Math. Soc., 11 (127), 112–117

[4] Heath-Brown D. R., “Fractional moments of the Riemann zeta-function”, J. London Math. Soc., 24(2):1 (1981), 65–78 | DOI | MR | Zbl

[5] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR

[6] Elliott P. D. T. A., “On the distribution of $\operatorname{arg}L(s,\chi)$ in the half-plane $\sigma>1/2$”, Acta arith, XX (1972), 155–169 | MR

[7] Laurinchikas A., “Raspredelenie znachenii kompleksnoznachnykh funktsii”, Lit. matem. sb., XV:2 (1975), 25–39 | MR