On the riemann zeta-function and the divisor problem II

• Autorzy: Aleksandar Ivić
• 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 $$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$ .

Słowa kluczowe

EN

Dirichlet divisor problem; Riemann zeta-function; power moments of$$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$; power moments of E*(t); 11N37; 11M06

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 2
• Strony: 203-214

Daty

wydano

2005-06-01

online

2005-06-01

Twórcy

autor

Aleksandar Ivić

• Universiteta u Beogradu

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.

Identyfikatory

DOI

10.2478/BF02479196