On number of zeros of the Riemann zeta function that lie in > very short intervals of neighborhood of the critical line

Do Duc Tam

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 106-124

Résumé

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.

Détail

Keywords: zeta function, non-trivial zeros, critical line.