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Abstrakty
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.
Słowa kluczowe
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
1029-1040
Opis fizyczny
Daty
wydano
2010-12-01
online
2010-10-30
Twórcy
autor
- Universitet u Beogradu, ivic@rgf.bg.ac.rs
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