On Gram's law in the theory of the Riemann zeta function
Izvestiya. Mathematics, Tome 76 (2012) no. 2, pp. 275-309 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove an assertion of Selberg concerning Gram's rule and the distribution of zeros of the Riemann zeta function. We also prove some equivalent assertions.
Keywords: Gram's law, Gram's rule, argument of the Riemann zeta function.
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M. A. Korolev. On Gram's law in the theory of the Riemann zeta function. Izvestiya. Mathematics, Tome 76 (2012) no. 2, pp. 275-309. http://geodesic.mathdoc.fr/item/IM2_2012_76_2_a2/

[1] 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 | DOI | MR | Zbl

[2] C. L. Siegel, “Contributions to the theory of the Dirichlet $L$-series and the Epstein zeta-functions”, Ann. of Math. (2), 44:2 (1943), 143–172 | DOI | MR | Zbl

[3] A. A. Karatsuba, M. A. Korolev, “The argument of the Riemann zeta function”, Russian Math. Surveys, 60:3 (2005), 433–488 | DOI | MR | Zbl

[4] T. Christ, J. Kalpokas, J. Steuding, “Neue Resultate über die Wertverteilung der Riemannschen Zetafunktion auf der kritischen Geraden”, Math. Semesterber., 57:2 (2010), 201–229 | DOI | MR | Zbl

[5] J. I. Hutchinson, “On the roots of the Riemann zeta function”, Trans. Amer. Math. Soc., 27:1 (1925), 49–60 | DOI | MR | Zbl

[6] E. C. Titchmarsh, “The zeros of the Riemann zeta-function”, Proc. Roy. Soc. London Ser. A, 151:873 (1935), 234–255 | DOI | Zbl

[7] T. S. Trudgian, Further results on Gram's law, D. Phil. Thesis, University of Oxford, 2009

[8] A. Selberg, “The zeta-function and the Riemann hypothesis”, C. R. Dixieme Congrès Math. Scandinaves 1946, Jul. Gjellerups Forlag, Copenhagen, 1947, 187–200 | MR | Zbl

[9] A. Selberg, “Contributions to the theory of the Riemann zeta-function”, Arch. Math. Naturvid., 48:5 (1946), 89–155 | MR | Zbl

[10] A. Ghosh, “On the Riemann zeta-function – mean value theorems and the distribution of $|S(t)|$”, J. Number Theory, 17:1 (1983), 93–102 | DOI | MR | Zbl

[11] A. Selberg, Collected papers, v. I, Springer-Verlag, Berlin, 1989 | MR | Zbl

[12] A. G. Postnikov, “Ergodic problems in the theory of congruences and of diophantine approximations”, Proc. Steklov Inst. Math., 82 (1966), 1–128 | MR | MR | Zbl | Zbl

[13] V. F. Gaposhkin, “On the speed of convergence to the Gaussian law of weighted sums of gap series”, Theory Probab. Appl., 13:3 (1968), 421–437 | DOI | MR | Zbl | Zbl

[14] A. Fujii, “On the zeros of Dirichlet $L$-functions. V”, Acta Arith., 28:4 (1976), 395–403 | MR | Zbl

[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 | DOI | MR | Zbl

[16] G. M. Fikhtengol'ts, Differential and integral calculus, v. III, Deutscher, Berlin, 1977 | MR | Zbl | Zbl

[17] A. Selberg, “Old and new conjectures and results about a class of Dirichlet series”, Proceedings of the Amalfi conference on analytic number theory (Maiori, Italy, 1989), Universitàt di Salerno, Salerno, 1992, 367–385 | MR | Zbl

[18] M. Loève, Probability theory, Van Nostrand, Toronto–New York–London, 1960 | MR | MR | Zbl

[19] D. A. Goldston, S. M. Gonek, “A note on $S(t)$ and the zeros of the Riemann zeta-function”, arXiv: math/0511092

[20] M. A. Korolev, “On multiple zeros of the Riemann zeta-function”, Izv. Math., 70:3 (2006), 427–446 | DOI | MR | Zbl