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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 3(363), Pages 41–96 (Mi umn1429)  

This article is cited in 33 scientific papers (total in 34 papers)

The argument of the Riemann zeta function

A. A. Karatsubaa, M. A. Korolev

a Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is a survey of the main results concerning the behaviour of the argument of the Riemann zeta function on the critical line.

DOI: https://doi.org/10.4213/rm1429

Full text: PDF file (531 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2005, 60:3, 433–488

Bibliographic databases:

UDC: 511
MSC: 11M06, 11M26
Received: 18.01.2005

Citation: A. A. Karatsuba, M. A. Korolev, “The argument of the Riemann zeta function”, Uspekhi Mat. Nauk, 60:3(363) (2005), 41–96; Russian Math. Surveys, 60:3 (2005), 433–488

Citation in format AMSBIB
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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 multiple zeros of the Riemann zeta function”, Izv. Math., 70:3 (2006), 427–446  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. 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
    3. Slater P.B., “Fractal fits to Riemann zeros”, Can. J. Phys., 85:4 (2007), 345–357  crossref  adsnasa  isi  elib
    4. Goldston D.A., Gonek S.M., “A note on $S(t)$ and the zeros of the Riemann zeta-function”, Bull. Lond. Math. Soc., 39:3 (2007), 482–486  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. M A. Korolev, “Selberg's conjecture concerning the distribution of imaginary parts of zeros of the Riemann zeta function”, Dokl. Math., 78:1 (2008), 531–534  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. Castro C., “On the Riemann hypothesis, area quantization, Dirac operators, modularity, and renormalization group”, Int. J. Geom. Methods Mod. Phys., 7:1 (2010), 1–31  crossref  mathscinet  zmath  isi  elib
    7. M. A. Korolev, “Gram's law and Selberg's conjecture on the distribution of zeros of the Riemann zeta function”, Izv. Math., 74:4 (2010), 743–780  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. R. N. Boyarinov, “On large distances between neighbouring zeros of the Riemann zeta-function”, Discrete Math. Appl., 20:4 (2010), 411–420  mathnet  crossref  crossref  mathscinet  elib  elib
    9. R. N. Boyarinov, “Argument dzeta-funktsii Rimana”, Chebyshevskii sb., 11:1 (2010), 54–67  mathnet  mathscinet
    10. R. N. Boyarinov, “On Large Values of the Function $S(t)$ on Short Intervals”, Math. Notes, 89:4 (2011), 472–479  mathnet  crossref  crossref  mathscinet  isi
    11. R. N. Boyarinov, “On the Zeros of the Riemann Zeta Function of Large Multiplicity”, Math. Notes, 89:5 (2011), 613–618  mathnet  crossref  crossref  mathscinet  isi
    12. M. A. Korolev, “Letter to the editors”, Izv. Math., 75:4 (2011), 869–869  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. 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
    14. M. A. Korolev, “On Gram's law in the theory of the Riemann zeta function”, Izv. Math., 76:2 (2012), 275–309  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. M. A. Korolev, “On Karatsuba's problem related to Gram's law”, Proc. Steklov Inst. Math., 276 (2012), 156–166  mathnet  crossref  mathscinet  isi  elib  elib
    16. R. N. Boyarinov, “On the number of Gram's intervals containing the ordinates of successive zeros of the Riemann zeta function”, Discrete Math. Appl., 22:5-6 (2012), 683–692  mathnet  crossref  crossref  mathscinet  elib
    17. M. A. Korolev, “On Selberg formulae related to Gram's law”, Sb. Math., 203:12 (2012), 1808–1816  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Emanuel Carneiro, Vorrapan Chandee, Micah B. Milinovich, “Bounding
      $$S(t)$$
      and
      $$S_1(t)$$
      on the Riemann hypothesis”, Math. Ann, 2012  crossref  mathscinet  isi
    19. Korolev M., “Gram's Law and the Argument of the Riemann Zeta Function”, Publ. Inst. Math.-Beograd, 92:106 (2012), 53–78  crossref  mathscinet  zmath  isi
    20. M. A. Korolev, “On small values of the Riemann zeta-function at Gram points”, Sb. Math., 205:1 (2014), 63–82  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Korolev M.A., “On Large Values of the Riemann Zeta-Function on Short Segments of the Critical Line”, Acta Arith., 166:4 (2014), 349–390  crossref  mathscinet  zmath  isi
    22. Anne-Maria Ernvall-Hytönen, Almasa Odžak, Lejla Smajlović, Medina Sušić, “On the modified Li criterion for a certain class of L−functions”, Journal of Number Theory, 2015  crossref  mathscinet  isi
    23. M. A. Korolev, “On the Horizontal Distribution of Zeros of the Functions $\operatorname{Re} \zeta(s)$ and $\operatorname{Im}\zeta(s)$”, Math. Notes, 98:6 (2015), 986–989  mathnet  crossref  crossref  mathscinet  isi  elib
    24. M. A. Korolev, “Gram's law in the theory of the Riemann zeta-function. Part 1”, Proc. Steklov Inst. Math., 292, suppl. 2 (2016), S1–S146  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    25. Carneiro E., Finder R., “on the Argument of l-Functions”, 46, no. 4, 2015, 601–620  crossref  mathscinet  zmath  isi
    26. M. A. Korolev, “Gram's Law in the Theory of Riemann Zeta-Function. Part 2”, Proc. Steklov Inst. Math., 294, suppl. 1 (2016), 1–78  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    27. Korolev M.A., “An extreme values of the function S(T) in short intervals”, Indian J. Pure Appl. Math., 47:4 (2016), 603–615  crossref  mathscinet  zmath  isi  scopus
    28. Odzak A., “On the asymptotic criterion for the zero-free regions of certain $L$-functions”, Turk. J. Math., 40:3 (2016), 688–702  crossref  mathscinet  isi  elib
    29. 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  mathscinet  isi  elib  elib
    30. Aschheim R., Perelman C.C., Irwin K., “The Search For a Hamiltonian Whose Energy Spectrum Coincides With the Riemann Zeta Zeroes”, Int. J. Geom. Methods Mod. Phys., 14:6 (2017)  crossref  mathscinet  zmath  isi
    31. Carneiro E., Chirre A., “Bounding S-N(T) on the Riemann Hypothesis”, Math. Proc. Camb. Philos. Soc., 164:2 (2018), 259–283  crossref  mathscinet  zmath  isi
    32. Perelman C.C., “On the Riemann Hypothesis, Complex Scalings and Logarithmic Time Reversal”, J. Geom. Phys., 129 (2018), 133–141  crossref  mathscinet  zmath  isi
    33. 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
    34. Carneiro E., Chirre A., Milinovich M.B., “Bandlimited Approximations and Estimates For the Riemann Zeta-Function”, Publ. Mat., 63:2 (2019), 601–661  crossref  isi
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