On the riemann zeta-function and the divisor problem

Aleksandar Ivić

Open Mathematics (2004)

  • Volume: 2, Issue: 4, page 494-508
  • 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 ς 1 2 + i t . If E * t = E t - 2 π Δ * t / 2 π with Δ * x = - Δ x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain ∫ 0 T E * t 4 d t ≪ e T 16 / 9 + ε . We also show how our method of proof yields the bound ∑ r = 1 R ∫ t r - G t r + G ς 1 2 + i t 2 d t 4 ≪ e T 2 + e G - 2 + R G 4 T ε , where T 1/5+ε≤G≪T, T

How to cite

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Aleksandar Ivić. "On the riemann zeta-function and the divisor problem." Open Mathematics 2.4 (2004): 494-508. <http://eudml.org/doc/268928>.

@article{AleksandarIvić2004,
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 \[\left| \{\varsigma \left( \{\tfrac\{1\}\{2\} + it\} \right)\} \right|\] . 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},
author = {Aleksandar Ivić},
journal = {Open Mathematics},
keywords = {11N37; 11M06},
language = {eng},
number = {4},
pages = {494-508},
title = {On the riemann zeta-function and the divisor problem},
url = {http://eudml.org/doc/268928},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Aleksandar Ivić
TI - On the riemann zeta-function and the divisor problem
JO - Open Mathematics
PY - 2004
VL - 2
IS - 4
SP - 494
EP - 508
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 \[\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|\] . 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
LA - eng
KW - 11N37; 11M06
UR - http://eudml.org/doc/268928
ER -

References

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