A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma < 1$

Yuk-Kam Lau

doi:10.5802/jtnb.553

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 \leq 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, Tome 18 (2006) no. 2, pp. 445-470.

Résumé
Abstract

Pour $σ$ dans la bandre critique $1 / 2 < σ < 1$ , on note $E_{σ} (T)$ le terme d’erreur de la formule asymptotique de $\int_{1}^{T} {| ζ (σ + i t) |}^{2} d t$ (pour $T$ grand). C’est un analogue du terme d’erreur classique $E (T)$ ( $= E_{1 / 2} (T)$ ). L’étude de $E (T)$ a une longue histoire, mais celle de $E_{σ} (T)$ est assez récente. En particulier, lorsque $3 / 4 < σ < 1$ , on connaît peu d’informations sur $E_{σ} (T)$ . Pour en gagner, nous étudions la moyenne $\int_{1}^{T} E_{σ} (u) d u$ . Dans cet article, nous donnons une expression en série de type Atkinson et explorons quelques une des propriétés de la moyenne comme fonction en $T$ .

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 $\int_{1}^{T} E_{σ} (u) d u$ . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$ .

MR Zbl

DOI : 10.5802/jtnb.553

Affiliations des auteurs :

Yuk-Kam Lau ¹

¹ Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong

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     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 $3/4\le \sigma < 1$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {445--470},
     publisher = {Universit\'e Bordeaux 1},
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Yuk-Kam Lau. A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma < 1$. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 445-470. doi : 10.5802/jtnb.553. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.553/

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