PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2004 | 2 | 4 | 494-508
Tytuł artykułu

On the riemann zeta-function and the divisor problem

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $$\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|$$ . If $$E^* \left( t \right) = E\left( t \right) - 2\pi \Delta ^* \left( {t / 2\pi } \right)$$ with $$\Delta ^* \left( x \right) = - \Delta \left( x \right) + 2\Delta \left( {2x} \right) - \tfrac{1}{2}\Delta \left( {4x} \right)$$ , then we obtain $$\int_0^T {\left( {E^* \left( t \right)} \right)^4 dt \ll _e T^{16/9 + \varepsilon } } $$ . We also show how our method of proof yields the bound $$\sum\limits_{r = 1}^R {\left( {\int_{tr - G}^{tr + G} {\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|^2 dt} } \right)^4 \ll _e T^{2 + e} G^{ - 2} + RG^4 T^\varepsilon } $$ , where T 1/5+ε≤G≪T, T
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
4
Strony
494-508
Opis fizyczny
Daty
wydano
2004-08-01
online
2004-08-01
Twórcy
Bibliografia
  • [1] F.V. Atkinson: “The mean value of the zeta-function on the critical line”, Quant. J. Math. Oxford, Vol. 10, (1939), pp. 122–128.
  • [2] F.V. Atkinson: “The mean value of the Riemann zeta-function”, Acta Math., Vol. 81, (1949), pp. 353–376. http://dx.doi.org/10.1007/BF02395027
  • [3] D.R. Heath-Brown: “The twelfth power moment of the Riemann zeta-function”, Quart. J. Math. (Oxford), Vol. 29, (1978), pp. 443–462.
  • [4] D.R. Heath-Brown: “The distribution of moments in the Dirichlet divisor problems”, Acta Arith., Vol. 60, (1992), pp. 389–415.
  • [5] M.N. Huxley: Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996.
  • [6] A. Ivić: “Large values of the error term in the divisor problem”, Invent. Math., Vol. 71, (1983), pp. 513–520. http://dx.doi.org/10.1007/BF02095990
  • [7] A. Ivić: The Riemann zeta-function, John Wiley & Sons, New York, 1985.
  • [8] A. Ivić: The mean values of the Riemann zeta-function, LNs 82, Tata Inst. of Fundamental Research, Bombay (distr. by Springer Verlag, Berlin etc.), 1991.
  • [9] A. Ivić: “Power moments of the Riemann zeta-function over short intervals”. Arch. Mat., Vol. 62, (1994), pp. 418–424. http://dx.doi.org/10.1007/BF01196431
  • [10] A. Ivić: On some problems involving the mean square of |ξ(1/2+it)|. Bull. CXVI Acad. Serbe, Classe des Sciences mathématique, Vol. 23, (1998), pp. 71–76.
  • [11] A. Ivić: “Sums of squares of |ξ(1/2+it)| over short intervals”, Max-Planck-Institut für Mathematik, Preprint Series, Vol. 52, (2002), pp. 12.
  • [12] A. Ivić and P. Sargos: On the higher moments of the error term in the divisor problem, to appear.
  • [13] M. Jutila: “Riemann’s zeta-function and the divisor problem”, Arkiv Mat., Vol. 21, (1983), pp. 75–96; ibid. Vol. 31, (1993), pp. 61–70. http://dx.doi.org/10.1007/BF02384301
  • [14] M. Jutila: “On a formula of Atkinson”, In: Proc. Coll. Soc. J. Bolyai-Budapest’81, Vol. 34, North-Holland, Amsterdam, 1984, pp. 807–823.
  • [15] K. Matsumoto: “Recent developments in the mean square theory of the Riemann zeta and other zeta-functions”, In: Number Theory, Birkhäuser, Basel, 2000, pp. 241–286.
  • [16] T. Meurman: “A generalization of Atkinson’s formula to L-functions”, Acta Arith., Vol. 47, (1986), pp. 351–370.
  • [17] O. Robert and P. Sargos: “Three-dimensional exponential sums with monomials”, J. reine angew. Math. (in print).
  • [18] E.C. Titchmarsh: The theory of the Riemann zeta-function, 2nd Ed., University Press, Oxford, 1986.
  • [19] K.-M. Tsang: “Higher power moments of Δ(x), E(t) and P(x)”, Proc. London Math. Soc. (3), Vol. 65, (1992), pp. 65–84.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475958
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.