Geodesic
On number of zeros of the Riemann zeta function that lie in > very short intervals of neighborhood of the critical line
Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 106-124 Cet article a éte moissonné depuis la source Math-Net.Ru

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Proof (or disproof) of the Riemann hypothesis is the central problem of analytic number theory. By now it has not been solved. In 1985 Karatsuba proved that for any $ 0 \varepsilon 0,001 $, $ 0,5 \sigma \leq 1 $, $ T> T_0 (\varepsilon)> 0 $ and $ H = T ^ { 27/82 + \varepsilon} $ in the rectangle with vertices $ \sigma + iT $, $ \sigma + i (T + H) $, $ 1 + i (T + H) $, $ 1 + iT $ contains no more than $ cH / (\sigma-0,5) $ zeros of $ \zeta (s) $. Thereby A.A. Karatsuba significantly strengthened the classical theorem J. Littlewood's. Decrease in magnitude of $H$ for individual rectangle has not been obtained. However, by solving this problem «on average», in 1989 L.V. Kiseleva proved that for «almost all» $ T $ in the interval $ [X, X + X ^ {11/12 + \varepsilon}] $, $ X> X_0 (\varepsilon) $ in rectangle with vertices $ \sigma + iT $, $ \sigma + i (T + X ^ \varepsilon) $, $ 1 + i (T + X ^ \varepsilon) $, $ 1 + iT $ contains no more than $ O (X ^ \varepsilon / (\sigma-0,5)) $ zeros of $ \zeta (s) $. In this article, we obtain a result of this kind, but for «almost all » $ T $ in the interval $ [X, X + X ^ {7/8 + \varepsilon}] $. Bibliography: 23 titles.
Keywords: zeta function, non-trivial zeros, critical line. 
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Do Duc Tam. On number of zeros of the Riemann zeta function that lie in <> very short intervals of neighborhood of the critical line. Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 106-124. http://geodesic.mathdoc.fr/item/CHEB_2016_17_3_a7/

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