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Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 3, Pages 569–584 (Mi izv1456)  

This article is cited in 21 scientific papers (total in 23 papers)

On the zeros of the function $\zeta(s)$ on short intervals of the critical line

A. A. Karatsuba


Abstract: It is proved that for any $\varepsilon>0$ there exists $c=c(\varepsilon)>0$ such that for $T\geqslant T_0(\varepsilon)>0$ and $H=T^{27/82+\varepsilon}$ we have $N_0(T+H)-N_0(T)\geqslant cH\ln T$, where $N_0(T)$ is the number of odd order zeros of $\zeta(\frac12+it)$ in the interval $(0,T)$.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 24:3, 523–537

Bibliographic databases:

UDC: 511
MSC: Primary 10H05; Secondary 10G10
Received: 17.02.1984

Citation: A. A. Karatsuba, “On the zeros of the function $\zeta(s)$ on short intervals of the critical line”, Izv. Akad. Nauk SSSR Ser. Mat., 48:3 (1984), 569–584; Math. USSR-Izv., 24:3 (1984), 523–537

Citation in format AMSBIB
\Bibitem{Kar84}
\by A.~A.~Karatsuba
\paper On~the zeros of the function $\zeta(s)$ on short intervals of the critical line
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1984
\vol 48
\issue 3
\pages 569--584
\mathnet{http://mi.mathnet.ru/izv1456}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=747251}
\zmath{https://zbmath.org/?q=an:0545.10026}
\transl
\jour Math. USSR-Izv.
\yr 1984
\vol 24
\issue 3
\pages 523--537
\crossref{https://doi.org/10.1070/IM1985v024n03ABEH001246}


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    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. A. A. Karatsuba, “The distribution of zeros of the function $\zeta(1/2+it)$”, Math. USSR-Izv., 25:3 (1985), 519–529  mathnet  crossref  mathscinet  zmath
    2. A. A. Karatsuba, “On the zeros of the function $\zeta(s)$ in the neighborhood of the critical line”, Math. USSR-Izv., 26:2 (1986), 307–313  mathnet  crossref  mathscinet  zmath
    3. A. A. Karatsuba, “On the real zeros of the function $\zeta(1/2+it)$”, Russian Math. Surveys, 40:4 (1985), 191–192  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. A. Karatsuba, “The Riemann zeta function and its zeros”, Russian Math. Surveys, 40:5 (1985), 23–82  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. L. V. Kiseleva, “On the number of zeros of the function $\zeta(s)$ on “almost all” short intervals of the critical line”, Math. USSR-Izv., 32:3 (1989), 475–499  mathnet  crossref  mathscinet  zmath
    6. A. A. Karatsuba, “On the zeros of the Davenport–Heilbronn function lying on the critical line”, Math. USSR-Izv., 36:2 (1991), 311–324  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. A. A. Karatsuba, “Zeros of certain Dirichlet series”, Russian Math. Surveys, 45:1 (1990), 207–208  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. A. A. Karatsuba, “The distribution of prime numbers”, Russian Math. Surveys, 45:5 (1990), 99–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. A. A. Karatsuba, “On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line”, Russian Acad. Sci. Izv. Math., 40:2 (1993), 353–376  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. S. A. Gritsenko, “Approximate functional equation for the product of two Dirichlet $L$-functions”, Russian Acad. Sci. Izv. Math., 45:2 (1995), 255–280  mathnet  crossref  mathscinet  zmath  isi
    11. A. A. Karatsuba, “Density theorem and the behavior of the argument of the Riemann zeta function”, Math. Notes, 60:3 (1996), 333–334  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. A. A. Karatsuba, “On the function $S(t)$”, Izv. Math., 60:5 (1996), 901–931  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. G. I. Arkhipov, V. I. Blagodatskikh, V. S. Vladimirov, S. M. Voronin, O. B. Lupanov, E. F. Mishchenko, Yu. V. Prokhorov, V. N. Chubarikov, A. B. Shidlovskii, “Anatolii Alekseevich Karatsuba (on his 60th birthday)”, Russian Math. Surveys, 53:2 (1998), 419–422  mathnet  crossref  crossref  mathscinet  zmath  isi
    14. A. A. Karatsuba, “Zeros and Local Extrema of Trigonometric Sums”, Problems Inform. Transmission, 39:1 (2003), 78–91  mathnet  crossref  mathscinet  zmath
    15. A. A. Karatsuba, M. A. Korolev, “Behaviour of the argument of the Riemann zeta function on the critical line”, Russian Math. Surveys, 61:3 (2006), 389–482  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. S. A. Gritsenko, E. A. Karatsuba, M. A. Korolev, I. S. Rezvyakova, D. I. Tolev, M. E. Changa, “Scientific Achievements of Anatolii Alekseevich Karatsuba”, Proc. Steklov Inst. Math., 280, suppl. 2 (2013), S1–S22  mathnet  crossref  crossref  zmath  isi  elib
    17. A. A. Karatsuba, “Venskii doklad: o kolichestve nulei dzeta-funktsii Rimana na korotkikh promezhutkakh kriticheskoi pryamoi”, Chebyshevskii sb., 16:1 (2015), 19–31  mathnet  elib
    18. Do Dyk Tam, “Raspredelenie nulei, lezhaschikh na kriticheskoi pryamoi, lineinykh kombinatsii $L$-funktsii Dirikhle”, Chebyshevskii sb., 16:3 (2015), 183–208  mathnet  elib
    19. Do Dyk Tam, “O nulyakh dzeta-funktsii Rimana $\zeta(s)$, lezhaschikh na pochti vsekh korotkikh promezhutkakh kriticheskoi pryamoi”, Chebyshevskii sb., 17:1 (2016), 71–89  mathnet  elib
    20. Do Dyk Tam, “O kolichestve nulei dzeta-funktsii Rimana, lezhaschikh v «pochti vsekh» ochen korotkikh promezhutkakh okrestnosti kriticheskoi pryamoi”, Chebyshevskii sb., 17:3 (2016), 106–124  mathnet  elib
    21. A. A. Karatsuba, “On zeros of certain Dirichlet series”, Proc. Steklov Inst. Math., 296, suppl. 2 (2017), 6–10  mathnet  crossref  crossref  isi  elib
    22. M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43  mathnet  crossref  crossref  isi  elib  elib
    23. Ivic A.P., Korolev M.A., “On the Distribution of Values of the Argument of the Riemann Zeta-Function”, J. Number Theory, 200 (2019), 96–131  crossref  mathscinet  zmath  isi  scopus
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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