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Tr. Mat. Inst. Steklova, 2002, Volume 239, Pages 215–238 (Mi tm369)  

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

The Argument of the Riemann Zeta Function on the Critical Line

M. A. Korolev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The following assertion is proved: If $N_1(T)$ is the number of sign changes of the argument of the Riemann zeta function $\zeta (s)$ on the interval $0<\operatorname{Im}s\le T$ of the critical line$\operatorname{Re}s=1/2$, then, for any $a$ such that $27/82<a\le 1$, $T\ge T_1(a)>0$, and $H=T^a$, the inequality $N_1(T+H)-N_1(T) \ge H\log T\exp (-\frac {c\log \log T}{\sqrt {\log \log \log T}})$ holds with a constant $c=c(a)>0$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 239, 202–224

Bibliographic databases:
UDC: 511
Received in May 2001

Citation: M. A. Korolev, “The Argument of the Riemann Zeta Function on the Critical Line”, Discrete geometry and geometry of numbers, Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov, Tr. Mat. Inst. Steklova, 239, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 215–238; Proc. Steklov Inst. Math., 239 (2002), 202–224

Citation in format AMSBIB
\Bibitem{Kor02}
\by M.~A.~Korolev
\paper The Argument of the Riemann Zeta Function on the Critical Line
\inbook Discrete geometry and geometry of numbers
\bookinfo Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 239
\pages 215--238
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm369}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1975145}
\zmath{https://zbmath.org/?q=an:1126.11332}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 239
\pages 202--224


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. A. Korolev, “On large values of the function $S(t)$ on short intervals”, Izv. Math., 69:1 (2005), 113–122  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. M. A. Korolev, “Sign changes of the function $S(t)$ on short intervals”, Izv. Math., 69:4 (2005), 719–731  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. A. Karatsuba, M. A. Korolev, “The argument of the Riemann zeta function”, Russian Math. Surveys, 60:3 (2005), 433–488  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. 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
    5. R. N. Boyarinov, “Argument dzeta-funktsii Rimana”, Chebyshevskii sb., 11:1 (2010), 54–67  mathnet  mathscinet
    6. R. N. Boyarinov, “Probabilistic methods in the theory of the Riemann zeta-function”, Theory Probab. Appl., 56:2 (2011), 181–192  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. Boyarinov R.N., “On the Value Distribution of the Riemann Zeta-Function”, Doklady Mathematics, 83:3 (2011), 290–292  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    8. 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
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