On the riemann zeta-function and the divisor problem

• 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 $$\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|$$ . 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

Słowa kluczowe

EN

Dirichlet divisor problem; Riemann zeta-function; mean square and twelfth moment of |ξ(1/2+it)|; mean fourth power of E*(t); 11N37; 11M06

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2004
• Tom: 2
• Numer: 4
• Strony: 494-508

Daty

wydano

2004-08-01

online

2004-08-01

Twórcy

autor

Aleksandar Ivić

• Universiteta u Beogradu

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.

Identyfikatory

DOI

10.2478/BF02475958