RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Chebyshevskii Sb., 2016, Volume 17, Issue 3, Pages 106–124 (Mi cheb500)  

This article is cited in 8 scientific papers (total in 8 papers)

On number of zeros of the Riemann zeta function that lie in «almost all» very short intervals of neighborhood of the critical line

Do Duc Tam

National Research University "Belgorod State University"

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}] $.
Bibliography: 23 titles.

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

Full text: PDF file (607 kB)
References: PDF file   HTML file
UDC: 511
Received: 11.06.2016
Accepted:13.09.2016

Citation: Do Duc Tam, “On number of zeros of the Riemann zeta function that lie in «almost all» very short intervals of neighborhood of the critical line”, Chebyshevskii Sb., 17:3 (2016), 106–124

Citation in format AMSBIB
\Bibitem{Do16}
\by Do~Duc~Tam
\paper On number of zeros of the Riemann zeta function that lie in <<almost all>> very short intervals of neighborhood of the critical line
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 3
\pages 106--124
\mathnet{http://mi.mathnet.ru/cheb500}
\elib{http://elibrary.ru/item.asp?id=27452085}


Linking options:
  • http://mi.mathnet.ru/eng/cheb500
  • http://mi.mathnet.ru/eng/cheb/v17/i3/p106

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. N. Kuznetsov, O. A. Matveeva, “Approksimatsionnyi podkhod v nekotorykh zadachakh teorii ryadov Dirikhle s multiplikativnymi koeffitsientami”, Chebyshevskii sb., 17:4 (2016), 124–131  mathnet  crossref  elib
    2. O. A. Matveeva, V. N. Kuznetsov, “K zadache analiticheskogo prodolzheniya ryadov Dirikhle s konechnoznachnymi koeffitsientami kak tselykh funktsii na kompleksnuyu ploskost”, Chebyshevskii sb., 18:4 (2017), 286–296  mathnet  crossref
    3. O. A. Matveeva, V. N. Kuznetsov, “Approksimatsionnye polinomy Dirikhle i nekotorye svoistva $L$-funktsii Dirikhle”, Chebyshevskii sb., 18:4 (2017), 297–305  mathnet  crossref
    4. V. N. Kuznetsov, O. A. Matveeva, “Granichnoe povedenie i zadacha analiticheskogo prodolzheniya odnogo klassa ryadov Dirikhle s multiplikativnymi koeffitsientami kak tselykh funktsii na kompleksnuyu ploskost”, Chebyshevskii sb., 19:1 (2018), 124–137  mathnet  crossref
    5. V. N. Kuznetsov, O. A. Matveeva, “Pochti periodicheskie funktsii i svoistvo universalnosti L-funktsii Dirikhle”, Chebyshevskii sb., 19:2 (2018), 368–376  mathnet  crossref
    6. V. N. Kuznetsov, O. A. Matveeva, “K odnoi zadache Yu. V. Linnika”, Chebyshevskii sb., 19:3 (2018), 202–209  mathnet  crossref  elib
    7. V. N. Kuznetsov, O. A. Matveeva, “K probleme obobschennykh kharakterov”, Chebyshevskii sb., 19:3 (2018), 210–218  mathnet  crossref  elib
    8. V. N. Kuznetsov, O. A. Matveeva, “Analog teoremy Daffina–Sheffera dlya odnogo klassa ryadov Dirikhle s konechnoznachnymi koeffitsientami”, Chebyshevskii sb., 19:4 (2018), 243–251  mathnet  crossref  elib
  • Number of views:
    This page:108
    Full text:39
    References:24

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020