ON NUMBER OF ZEROS OF THE RIEMANN ZETA FUNCTION THAT LIE IN «ALMOST ALL» VERY SHORT INTERVALS OF NEIGHBORHOOD OF THE CRITICAL LINE
https://doi.org/10.22405/2226-8383-2016-17-3-106-124
Abstract
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 \( [X, X + X ^ {7/8 + \varepsilon}] \).
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Review
For citations:
Tam D. ON NUMBER OF ZEROS OF THE RIEMANN ZETA FUNCTION THAT LIE IN «ALMOST ALL» VERY SHORT INTERVALS OF NEIGHBORHOOD OF THE CRITICAL LINE. Chebyshevskii Sbornik. 2016;17(3):106-124. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-3-106-124
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