Gram's law and Selberg's conjecture on the distribution of zeros of the Riemann zeta function

M. A. Korolev

Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 743-780.

Résumé

We prove a number of new assertions related to the zeros of the Riemann zeta function $\zeta(s)$ and to the so-called Gram law.

Détail

Keywords: Riemann zeta function, Gram's law, Gram's rule, the argument of the Riemann zeta function.

Bibliographie

[1] J.-P. Gram, “Note sur les zéros de la fonction $\zeta(s)$ de Riemann”, Acta Math., 27:1 (1903), 289–304 | DOI | MR | Zbl

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

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

[4] E. C. Titchmarsh, “The zeros of the Riemann zeta-function”, Proc. Roy. Soc. London Ser. A, 157 (1936), 261–263 | DOI | Zbl

[5] A. Selberg, “The zeta-function and the Riemann hypothesis”, C. R. Dixième Congrès Math. Skandinaves 1946, 10, Jul. Gjellerups Forlag, Copenhagen, 1947, 187–200 | MR

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

[7] 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

[8] 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

[9] E. C. Titchmarsh, “The mean-value of the zeta-function on the critical line”, Proc. London Math. Soc. (2), 27:1 (1928), 137–150 | DOI | Zbl

[10] Ya. Mozer, “O zakone Grama v teorii dzeta-funktsii Rimana”, Acta Arith., 32:2 (1977), 107–113 | MR | Zbl

[11] Ya. Mozer, “Dokazatelstvo gipotezy E. K. Titchmarsha v teorii dzeta-funktsii Rimana”, Acta Arith., 36:2 (1980), 147–156 | MR | Zbl

[12] Ya. Mozer, “Ob odnoi bikvadratnoi summe v teorii dzeta-funktsii Rimana”, Acta Math. Univ. Comenian., 42–43 (1983), 35–39 | MR | Zbl

[13] A. Fujii, “A comparative study of the zeros of Dirichlet $L$-functions”, Bull. Amer. Math. Soc., 81 (1975), 199–201 | DOI | MR | Zbl

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

[15] A. Fujii, “Gram's law for the zeta zeros and the eigenvalues of Gaussian unitary ensembles”, Proc. Japan Acad. Ser. A Math. Sci., 63:10 (1987), 392–395 | DOI | MR | Zbl

[16] A. A. Lavrik, “Titchmarsh's problem in the discrete theory of the Riemann zeta-function”, Proc. Steklov Inst. Math., 207:6 (1995), 179–209 | MR | Zbl

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

[18] A. Ghosh, “On Riemann's zeta-function – sign changes of $S(T)$”, Recent progress in analytic number theory. Vol. I (Durham, 1979), Academic Press, London–New York, 1981, 25–46 | MR | Zbl

[19] A. A. Karatsuba, “On the function $S(t)$”, Izv. Math., 60:5 (1996), 901–931 | DOI | MR | Zbl

[20] M. A. Korolev, “The argument of the Riemann zeta-function on the critical line”, Izv. Math., 67:2 (2003), 225–264 | DOI | MR | Zbl

[21] A. A. Karatsuba, S. M. Voronin, The Riemann zeta-function, de Gruyter Exp. Math., 5, de Gruyter, Berlin, 1992 | MR | MR | Zbl | Zbl

[22] I. M. Vinogradov, Trigonometrical sums in number theory, Statistical Publ. Society, Calcutta, 1978 | MR | MR | Zbl | Zbl

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