A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip 3 / 4 ≤ σ < 1

Yuk-Kam Lau[1]

  • [1] Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 445-470
  • ISSN: 1246-7405

Abstract

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Let E σ ( T ) be the error term in the mean square formula of the Riemann zeta-function in the critical strip 1 / 2 < σ < 1 . It is an analogue of the classical error term E ( T ) . The research of E ( T ) has a long history but the investigation of E σ ( T ) is quite new. In particular there is only a few information known about E σ ( T ) for 3 / 4 < σ < 1 . As an exploration, we study its mean value ∫ 1 T E σ ( u ) d u . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of T .

How to cite

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Lau, Yuk-Kam. "A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma &lt; 1$." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 445-470. <http://eudml.org/doc/249650>.

@article{Lau2006,
abstract = {Let $E_\sigma (T)$ be the error term in the mean square formula of the Riemann zeta-function in the critical strip $1/2&lt;\sigma &lt;1$. It is an analogue of the classical error term $E(T)$. The research of $E(T)$ has a long history but the investigation of $E_\sigma (T)$ is quite new. In particular there is only a few information known about $E_\sigma (T)$ for $3/4&lt;\sigma &lt;1$. As an exploration, we study its mean value $\int _1^TE_\sigma (u)\,du$. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$.},
affiliation = {Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong},
author = {Lau, Yuk-Kam},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {445-470},
publisher = {Université Bordeaux 1},
title = {A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma &lt; 1$},
url = {http://eudml.org/doc/249650},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Lau, Yuk-Kam
TI - A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma &lt; 1$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 445
EP - 470
AB - Let $E_\sigma (T)$ be the error term in the mean square formula of the Riemann zeta-function in the critical strip $1/2&lt;\sigma &lt;1$. It is an analogue of the classical error term $E(T)$. The research of $E(T)$ has a long history but the investigation of $E_\sigma (T)$ is quite new. In particular there is only a few information known about $E_\sigma (T)$ for $3/4&lt;\sigma &lt;1$. As an exploration, we study its mean value $\int _1^TE_\sigma (u)\,du$. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$.
LA - eng
UR - http://eudml.org/doc/249650
ER -

References

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