On the riemann zeta-function and the divisor problem II

Aleksandar Ivić

Open Mathematics (2005)

  • Volume: 3, Issue: 2, page 203-214
  • ISSN: 2391-5455

Abstract

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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 ζ 1 2 + i t . If E *(t)=E(t)-2πΔ*(t/2π) with Δ * x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain ∫ 0 T E * t 5 d t ≪ ε T 2 + ε and ∫ 0 T E * t 544 75 d t ≪ ε T 601 225 + ε . It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of ζ 1 2 + i t .

How to cite

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Aleksandar Ivić. "On the riemann zeta-function and the divisor problem II." Open Mathematics 3.2 (2005): 203-214. <http://eudml.org/doc/268775>.

@article{AleksandarIvić2005,
abstract = {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 \[\left| \{\zeta \left( \{\frac\{1\}\{2\} + it\} \right)\} \right|\] .},
author = {Aleksandar Ivić},
journal = {Open Mathematics},
keywords = {11N37; 11M06},
language = {eng},
number = {2},
pages = {203-214},
title = {On the riemann zeta-function and the divisor problem II},
url = {http://eudml.org/doc/268775},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Aleksandar Ivić
TI - On the riemann zeta-function and the divisor problem II
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 203
EP - 214
AB - 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 \[\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|\] .
LA - eng
KW - 11N37; 11M06
UR - http://eudml.org/doc/268775
ER -

References

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  1. [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 Zbl0036.18603
  2. [2] D.R. Heath-Brown: “The twelfth power moment of the Riemann zeta-function”, Quart. J. Math. (Oxford), Vol. 29, (1978), pp. 443–462. Zbl0394.10020
  3. [3] M.N. Huxley: Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996. 
  4. [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 Zbl1065.11079
  5. [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 Zbl0489.10045
  6. [6] A. Ivić: The Riemann zeta-function, John Wiley & Sons, New York, 1985; 2nd Ed., Dover, Mineola, New York, 2003. 
  7. [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. [8] A. Ivić: “On the Riemann zeta-function and the divisor problem”, Central European J. Math., Vol. 2 (4), (2004), pp. 1–15. 
  9. [9] A. Ivić and P. Sargos: “On the higher moments of the error term in the divisor problem”, to appear. Zbl1234.11129
  10. [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 Zbl0513.10040
  11. [11] O. Robert and P. Sargos: “Three-dimensional exponential sums with monomials”, J. reine angew. Math., in press. Zbl1165.11067

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