# On the riemann zeta-function and the divisor problem

Open Mathematics (2004)

• Volume: 2, Issue: 4, page 494-508
• ISSN: 2391-5455

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## 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 $\left|\varsigma \left(\frac{1}{2}+it\right)\right|$ . If ${E}^{*}\left(t\right)=E\left(t\right)-2\pi {\Delta }^{*}\left(t/2\pi \right)$ with ${\Delta }^{*}\left(x\right)=-\Delta \left(x\right)+2\Delta \left(2x\right)-\frac{1}{2}\Delta \left(4x\right)$ , then we obtain ${\int }_{0}^{T}{\left({E}^{*}\left(t\right)\right)}^{4}dt{\ll }_{e}{T}^{16/9+\epsilon }$ . We also show how our method of proof yields the bound $\sum _{r=1}^{R}{\left({\int }_{tr-G}^{tr+G}{\left|\varsigma \left(\frac{1}{2}+it\right)\right|}^{2}dt\right)}^{4}{\ll }_{e}{T}^{2+e}{G}^{-2}+R{G}^{4}{T}^{\epsilon }$ , 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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15. [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. Zbl0959.11036
16. [16] T. Meurman: “A generalization of Atkinson’s formula to L-functions”, Acta Arith., Vol. 47, (1986), pp. 351–370. Zbl0561.10019
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