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2010 | 8 | 6 | 1029-1040
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On some problems involving Hardy’s function

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Języki publikacji
EN
Abstrakty
EN
Some problems involving the classical Hardy function $$ Z\left( t \right) = \zeta \left( {\frac{1} {2} + it} \right)\left( {\chi \left( {\frac{1} {2} + it} \right)} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right) $$, are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
8
Numer
6
Strony
1029-1040
Opis fizyczny
Daty
wydano
2010-12-01
online
2010-10-30
Twórcy
Bibliografia
  • [1] Feng S., Zeros of the Riemann zeta function on the critical line, preprint available at http://arxiv.org/abs/1003.0059
  • [2] Hafner J.L., Ivić A., On some mean value results for the Riemann zeta-function, Théorie des nombres, Quebec, 1987, de Gruyter, Berlin-New York, 1989, 348–358
  • [3] Hafner J.L., Ivić A., On the mean square of the Riemann zeta-function on the critical line, J. Number Theory, 1989, 32(2), 151–191 http://dx.doi.org/10.1016/0022-314X(89)90024-3
  • [4] Heath-Brown D.R., The distribution and moments of the error term in the Dirichlet divisor problems, Acta Arith., 1992, 60(4), 389–415
  • [5] Heath-Brown D.R., Tsang K., Sign changes of E(T), Δ(x), and P(x), J. Number Theory, 1994, 49(1), 73–83 http://dx.doi.org/10.1006/jnth.1994.1081
  • [6] Hejhal D.A., On a result of Selberg concerning zeros of linear combinations of L-functions, Internat. Math. Res. Notices, 2000, 11, 551–577 http://dx.doi.org/10.1155/S1073792800000301
  • [7] Ivić A., The Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985
  • [8] Ivić A., On sums of gaps between the zeros of ζ(s) on the critical line, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 1995, 6, 55–62
  • [9] Ivić A., On small values of the Riemann zeta-function on the critical line and gaps between zeros, Liet. Mat. Rink., 2002, 42(1), 31–45
  • [10] Ivić A., On the integral of Hardy’s function, Arch. Math. (Basel), 2004, 83(1), 41–47
  • [11] Ivić A., On the mean square of the divisor function in short intervals, J. Théor. Nombres Bordeaux, 2009, 21(2), 251–261
  • [12] Ivić A., On the Mellin transforms of powers of Hardy’s function, Hardy-Ramanujan J., 2010, 33, 32–58
  • [13] Jutila M., Atkinson’s formula for Hardy’s function, J. Number Theory, 2009, 129(11), 2853–2878 http://dx.doi.org/10.1016/j.jnt.2009.02.011
  • [14] Jutila M., An asymptotic formula for the primitive of Hardy’s function, Ark. Mat., DOI: 10.1007/s11512-010-0122-4
  • [15] Kalpokas J., Steuding J., On the value distribution of the Riemann zeta-function on the critical line, preprint available at http://arxiv.org/abs/0907.1910
  • [16] Keating J.P., Snaith N.C., Random matrix theory and L-functions at s = 1/2, Comm. Math. Phys., 2000, 214(1), 91–110 http://dx.doi.org/10.1007/s002200000262
  • [17] Korolëv M.A., On the integral of the Hardy function Z(t), Izv. Math., 2008, 72(3), 429–478 http://dx.doi.org/10.1070/IM2008v072n03ABEH002407
  • [18] Montgomery H.L., The pair correlation of zeros of the zeta-function, In: Analytic number theory, St. Louis, 1972, Proc. Sympos. Pure Math., 24, AMS, Providence, 1973, 181–193
  • [19] Odlyzko A.M., On the distribution of spacings between zeros of the zeta function, Math. Comp., 1987, 48(177), 273–308
  • [20] Odlyzko A.M., The 1020-th zero of the Riemann zeta-function and 175 million of its neighbors, preprint available at http://www.dtc.umn.edu/sodlyzko/unpublished/zeta.10to20.1992.pdf
  • [21] Ramachandra K., On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, Tata Inst. Fund. Res. Lectures on Math. and Phys., 85, Tata Institute of Fundamental Research, Bombay, 1995
  • [22] Selberg A., Collected Papers. Vol. 1, Springer, Berlin, 1989
  • [23] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, New York, 1986
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-010-0071-y
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