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2005 | 3 | 2 | 203-214
Tytuł artykułu

On the riemann zeta-function and the divisor problem II

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 $$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$ . If E *(t)=E(t)-2πΔ*(t/2π) with $$\Delta *\left( x \right) + 2\Delta \left( {2x} \right) - \frac{1}{2}\Delta \left( {4x} \right)$$ , then we obtain $$\int_0^T {\left| {E*\left( t \right)} \right|^5 dt} \ll _\varepsilon T^{2 + \varepsilon } $$ and $$\int_0^T {\left| {E*\left( t \right)} \right|^{\frac{{544}}{{75}}} dt} \ll _\varepsilon T^{\frac{{601}}{{225}} + \varepsilon } .$$ It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of $$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$ .
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
2
Strony
203-214
Opis fizyczny
Daty
wydano
2005-06-01
online
2005-06-01
Twórcy
Bibliografia
  • [1] 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
  • [2] D.R. Heath-Brown: “The twelfth power moment of the Riemann zeta-function”, Quart. J. Math. (Oxford), Vol. 29, (1978), pp. 443–462.
  • [3] M.N. Huxley: Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996.
  • [4] M.N. Huxley: “Exponential sums and lattice points III”, Proc. London Math. Soc., Vol. 387, (2003), pp. 591–609. http://dx.doi.org/10.1112/S0024611503014485
  • [5] 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
  • [6] A. Ivić: The Riemann zeta-function, John Wiley & Sons, New York, 1985; 2nd Ed., Dover, Mineola, New York, 2003.
  • [7] A. Ivić: The mean values of the Riemann zeta-function, LNs, Vol. 82, Tata Inst. of Fundamental Research, Bombay (distr. by Springer Verlag, Berlin etc.), 1991.
  • [8] A. Ivić: “On the Riemann zeta-function and the divisor problem”, Central European J. Math., Vol. 2 (4), (2004), pp. 1–15.
  • [9] A. Ivić and P. Sargos: “On the higher moments of the error term in the divisor problem”, to appear.
  • [10] M. Jutila: “Riemann's zeta-function and the divisor problem”, Arkiv Mat., Vol. 21, (1983), pp. 75–96; “Riemann's zeta-function and the divisor problem II”, Arkiv Mat., Vol. 31, (1993), pp. 61–70. http://dx.doi.org/10.1007/BF02384301
  • [11] O. Robert and P. Sargos: “Three-dimensional exponential sums with monomials”, J. reine angew. Math., in press.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02479196
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